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ESCANABA –Bay College Campus Activities Board announces Comedian Ryan Conner will be appearing at 7 p.m. (Eastern time) in the Besse Theater on Wednesday.
Tickets are on sale now, and are just $1 for all students as well as $5 general admission.
Originally from Virginia, Conner comes from a large multi-racial family of 11 brothers.
A 12-year veteran of the comedy scene, Campus Activities Board is excited to bring him to campus.
This show is part of the Campus Activities Board schedule of events.
More information on CAB events can be found at http:// www.baycollege.edu/cab and https://www.facebook.com/baycab.


text:
This article is about the particular significance of the year 1874 to Wales and its people.

Incumbents

Lord Lieutenant of Anglesey – William Owen Stanley
Lord Lieutenant of Brecknockshire – Charles Morgan, 1st Baron Tredegar
Lord Lieutenant of Caernarvonshire – Edward Douglas-Pennant, 1st Baron Penrhyn
Lord Lieutenant of Cardiganshire – Edward Pryse
Lord Lieutenant of Carmarthenshire – John Campbell, 2nd Earl Cawdor
Lord Lieutenant of Denbighshire – William Cornwallis-West
Lord Lieutenant of Flintshire – Sir Stephen Glynne, 9th Baronet (until 17 June) Hugh Robert Hughes (from 4 August)
Lord Lieutenant of Glamorgan – Christopher Rice Mansel Talbot
Lord Lieutenant of Merionethshire – Edward Lloyd-Mostyn, 2nd Baron Mostyn
Lord Lieutenant of Monmouthshire – Henry Somerset, 8th Duke of Beaufort
Lord Lieutenant of Montgomeryshire – Sudeley Hanbury-Tracy, 3rd Baron Sudeley
Lord Lieutenant of Pembrokeshire – William Edwardes, 4th Baron Kensington
Lord Lieutenant of Radnorshire – John Walsh, 1st Baron Ormathwaite

Bishop of Bangor – James Colquhoun Campbell
Bishop of Llandaff – Alfred Ollivant
Bishop of St Asaph – Joshua Hughes
Bishop of St Davids – Connop Thirlwall (until May); Basil Jones (from 24 August)

Events
24 January – Four pilots and two apprentices are drowned in an accident off Llanddwyn on Anglesey.
17 February – In the 1874 United Kingdom general election, newly elected MPs include David Davies (Llandinam) at Cardigan (returned unopposed).
1 April – Frances Morgan marries Dr George Hoggan. They later open the first husband-and-wife general medical practice in the UK.
1 May – Coal-owner Sir George Elliot is raised to the baronetcy by the new prime minister, Disraeli.
15 July – Laying of the foundation stone of the clock tower at Machynlleth, built to mark the coming of age in 1873 of Viscount Castlereagh, the eldest son of the 5th Marquess of Londonderry of Plas Machynlleth.
20 July – In a mining accident at Charles Colliery, Llansamlet, nineteen men are killed.
October – The Western Mail reports a deathbed confession made to a minister in the US by a man who claimed he carried out the assault for which Dic Penderyn was executed in 1831.
24 August – William Basil Jones is consecrated Bishop of St David's.
5 October – The Powysland Club holds its annual meeting at the new Powysland Museum in Welshpool.
16 October – The first issue of Yr Ymwelydd is published in Australia under the editorship of William Meirion Evans.
unknown dates
Strike at Dinorwig slate quarry.
The Welsh Flannel Company is established at Holywell.
Henry Davis Pochin begins laying out Bodnant Garden.
John Mathias Berry and his wife Mary move to Merthyr Tydfil. The three sons born to them here go on to achieve success in business and be raised to the peerage: Henry Seymour Berry, 1st Baron Buckland, William Ewart Berry, 1st Viscount Camrose and Gomer Berry, 1st Viscount Kemsley.
Rev. Richard Williams Morgan is consecrated First Patriarch of a restored Ancient British Church by Jules Ferrette, the founder of the British Orthodox Church, taking the religious name of 'Mar Pelagius I' and undertaking to revive Celtic Christianity as practised prior to the Synod of Whitby while continuing duties as an Anglican clergyman.

Arts and literature
William Goscombe John begins assisting his father, a wood-carver, in his work at Cardiff Castle.

Awards
Evan Jones (Gurnos) wins the bardic chair at the Bangor eisteddfod.
William Thomas (Islwyn) wins a bardic chair at Rhyl.

New books
Sir William Boyd Dawkins – Cave Hunting
Gwaith Thomas Edwards (ed. Isaac Foulkes)
Isaac Foulkes – Rheinallt ab Gruffydd

Music
Joseph Parry becomes head of the new music department at University of Wales, Aberystwyth.
The London Welsh Choral Union presents Sarah Edith Wynne with a bust of herself by Joseph Edwards.

Sport
Rugby union
The first match is played at Cardiff Arms Park, between the Wanderers Club and Glamorgan 2nd XV.
Newport RFC and Swansea RFC are founded.

Births
4 February – Thomas Fagan Wallace, agricultural scientist (died 1951)
6 February – David Evans, composer (died 1948)
6 May – Miriam Kate Williams ("Vulcana"), strongwoman (died 1946)
20 May (in India) – Sir George Lewis Barstow, civil servant (died 1966)
2 July – George Boots — international rugby player (died 1928)
15 July – Gwyn Nicholls, Wales rugby captain (died 1939)
16 July – William Alexander Wales international rugby player (died 1937)
20 July – David Bowen (Myfyr Hefin), minister and author (died 1955)
30 July – Billy Meredith, footballer (died 1958)
31 October – James Henry Thomas, politician (died 1949)
12 November – Rachel Barrett, editor and suffragette (died 1953)
1 December – Dick Hellings, Wales international rugby player (died 1938)
30 December – Nantlais Williams, poet and religious leader (died 1959)
date unknown
William Eames, journalist (died 1958)
(in USA) Thomas David Edwards, composer (died 1930)

Deaths
3 January – Morris Williams (Nicander), author, 64
19 January – John Parry, editor, 61
19 April – Owen Jones, architect, 65
8 May (in Launceston, Tasmania) – Zephaniah Williams, Chartist leader, 78
17 June – Sir Stephen Glynne, 9th Baronet, 66
10 August – David Davies (Dai'r Cantwr), Rebecca rioter (born about 1812)
19 August – Joseph Kenny Meadows, illustrator, 83
3 October – Owen Williams (Owen Gwyrfai), poet, 84
14 November – John Ambrose Lloyd, musician, 59
19 November – Mary Pendrill Llewelyn, writer and translator (born 1811)

References

Wales

redpajama_set_name: RedPajamaWikipedia


text:
De plaats Angerlo telt 23 inschrijvingen in het rijksmonumentenregister. Hieronder een overzicht.

|}

Zie ook
 Lijst van rijksmonumenten in Zevenaar (gemeente) voor andere plaatsen in de gemeente.
 Lijst van gemeentelijke monumenten in Angerlo

Angerlo


text:
The Middle: Season 1
Back to DVD
Format: DVD Box Set
Certification: 15
Category: Television
Released: 26th September 2011
All episodes from the first season of the US sitcom following Frankie Heck (Patricia Heaton) and her family in the fictional town of Orson, Indiana. A devoted wife to quarry manager Mike (Neil Flynn) and mother to their three children, Axl (Charlie McDermott), Sue (Eden Sher) and Brick (Atticus Shaffer), Frankie spends her days balancing her job as a car salesperson with family life. Episodes comprise: 'Pilot', 'The Cheerleader', 'The Floating Anniversary', 'The Trip', 'The Block Party', 'The Front Door', 'The Scratch', 'Thanksgiving', 'Siblings', 'Christmas', 'The Jeans', 'The Neighbor', 'The Interview', 'The Yelling', 'Valentine's Day', 'The Bee', 'The Break-Up', 'The Fun House', 'The Final Four', 'TV Or Not TV', 'Worry Duty', 'Mother's Day', 'Signals' and 'Average Rules'.
Actors Charlie McDermott, Brian Doyle-Murray, Neil Flynn, Atticus Shaffer, Jeanette Miller, Eden Sher, Beau Wirick, Blaine Saunders, Patricia Heaton, Chris Kattan
Additional info DVD 9, Language(s): English, Castillian Spanish, French, Hard of Hearing Subtitles: English, Subtitles: English, Castillian Spanish, Danish, Finnish, French, Norwegian, Portuguese, Swedish, Interactive Menu, Scene Access, Screen ratio 1:1.78, Dolby Digital 5.1, Dolby Digital 2.0
Duration 497 mins

redpajama_set_name: RedPajamaCommonCrawl


text:
Share this Story: Students to sing national anthem in Ojibwa at Winnipeg Jets hockey game
PMN News
PMN Canada
Students to sing national anthem in Ojibwa at Winnipeg Jets hockey game
A redesigned Winnipeg Jets jersey in honour of Manitoba's Indigenous communities and cultures is shown in this undated handout photo. Singing the national anthem before a hockey game is a longstanding tradition, but at this Friday's Winnipeg Jets game, it will be sung in a non-traditional way. A group of Winnipeg students at Riverbend Community School will belt out the anthem in Ojibwe. Photo by HO /THE CANADIAN PRESS
WINNIPEG — The national anthem before this Friday's NHL game in Winnipeg won't quite sound the way it usually does.
A group of students from the city's Riverbend Community School are set to perform O Canada in Ojibwa.
Students to sing national anthem in Ojibwa at Winnipeg Jets hockey game Back to video
Gloria Barker, who teaches at the English-Ojibwa bilingual school, says it's a real honour to be able to promote the Indigenous language.
The Winnipeg Jets will also be sporting new jerseys with Indigenous logos during the pre-skate for their game against the Tampa Bay Lightning.
It's all part of a night to acknowledge the Winnipeg Aboriginal Sport Achievement Centre.
Barker says she is proud of her students and hopes they will inspire other Indigenous people and groups to support their traditional languages.
"Language is a very important part of who we are and our identity." (CTV Winnipeg)
This report by The Canadian Press was first published Jan. 15, 2020


baker's favorite knitting stitch markers.
Oh don't you just LOVE the aroma of fresh baking? These markers are all about the culinary arts! This set includes all the essentials!
Hand crank egg beaters, a "wooden" spoon and cookie spatula, a stand mixer, a measuring cup, a cook book, a stick of butter and a box of sugar on a progress keeper clip! Sweet little heart accent beadstoo!
My stitch markers are made with nickel-free/lead-free soldered rings so there are no ends to catch on delicate fibers. These rings will fit needles up to 4.5mm (US size 6), perfect for DK or worsted weight yarn knitting! Set of 7.


DalLastic outdoor rubber floor tiles | Dalhaus Dalsouple,DalLastic outdoor rubber flooring is designed for use around swimming pools, on balconies and roof gardens, decks, terraces, paths and walkways. DalLastic.
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Top 5 Tips for Exterior Tile Installations and Balconies - YouTube,30 Apr 2015 . www.custombuildingproducts Installing ceramic tile or stone on exterior balconies, patios, walkways and walls requires extra.
Balcony and Patio Flooring using Resitrix EPDM Rubber Membrane .,It's also used as the waterproof layer when installing a balcony floor. . We recommend tiling, paving or timber decking over our rubber membrane systems to.
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Balcony Floor Tiles - Alibaba,Balcony Floor Tiles, Wholesale Various High Quality Balcony Floor Tiles Products from Global Balcony Floor Tiles Suppliers and Balcony Floor Tiles Factory.
Interlocking composite decking tiles from Dura Composites - Dura .,Dura Tile composite decking tiles are interlocking and require no additional fixings, . DIY garden decking projects or for difficult to reach roof terraces and balconies. . a smooth, safe, angled transition between the tile and the floor surface.


Did you know that only about 1 in 10 million battery explosions can happen? While it's still a low number, we want you to avoid such a tragedy altogether. Vaping should be a series of fun and relaxing sessions for you. It shouldn't be a danger to your safety. Having said that, we have come up with some safety rules you should follow in regards to your batteries!
Use batteries that are used of lithium. They are efficient in storing power, but they do have the possibility to explode. Still, as mentioned above, it is very rare with only 1 in 10 million in occurrence (according to Battery University).
Do's: Your vapor should come with a charging unit which is designed to work well with your batteries. In the case that it malfunctions, do not purchase a generic one as it can do more damage than good. Instead, stick to the brand's manufacturers.
Dont's: Don't ever use batteries that are damaged. You'll know when they are if they are corroding or parts of them are destroyed.
Do's: Only charge your batteries on surfaces that are hard. Carpet and other linen materials can ignite a fire.
Don'ts: Do not overtighten your battery to the clearomiser as it can build some unnecessary pressure.
Do's: To save battery life, always turn it off when it is not in use. It keeps it durable, and it's beneficial for when you need it the most.
Don'ts: Just like any electronics, do not leave your batteries unattended if they're charging.
A vape shop like us wants you to continue enjoying a vape while practicing safety. Even though battery explosions happen, they are very very rare. Still, it's always better to be safe than sorry.
Super Awesome Smoke Tricks You Need to Learn Now!


To increase; to augment; to come to by way of increase; to be added as an increase, profit, or damage. Acquired; falling due; made or executed; matured; occurred; received; vested; was created; was incurred.
To attach itself to, as a subordinate or accessory claim or demand arises out of, and is joined to, its principal.
The term is also used of independent or original demands, meaning to arise, to happen, to come into force or existence; to vest, as in the phrase, "The right of action did not accrue within six years." To become a present right or demand; to come to pass.
Interest on money that a depositor has in a bank savings account accrues, so that after a certain time the amount will be increased by the amount of interest it has earned.
A Cause of Action, the facts that give a person a right to judicial relief, usually accrues on the date that the injury to the plaintiff is sustained. When the injury is not readily discoverable, the cause of action accrues when the plaintiff in fact discovers the injury. This occurs frequently in cases of Fraud or Malpractice. A woman, for example, has an appendectomy. Three years after the surgery, she still experiences dull pain on her right side. She is examined by another physician who discovers a piece of surgical sponge near the area of the operation. Although the injury had occurred at the time of surgery three years earlier, in this case the cause of action for Medical Malpractice accrues on the date that the sponge is discovered by the second doctor. This distinction is important for purposes of the running of the Statute of Limitations, the time set by law within which a lawsuit must be commenced after a cause of action accrues. In cases involving injuries that cannot be readily discovered, it would be unfair to bar a plaintiff from bringing a lawsuit because he or she does not start the suit within the required time from the date of injury.
v. 1) growing or adding to, such as interest on a debt or investment which continues to accumulate. 2) the coming into being of the right to bring a lawsuit. For example, the right to sue on a contract only accrues when the contract is breached (not mere suspicion that it might be breached) or when the other party repudiates the contract (anticipatory breach).
Foreign phrases: Confirmare nemo potest prius quam jus ei acciderit.No one can confirm a right before the right accrues to him.
(of a right or demand) to become capable of being enforced.
2.-To accrue means also to arise, to happen, to come to pass; as the statute of limitations does not commence running until the cause of action has accrued. 1 Bouv. Inst. n. 861; 2 Rawle, 277; 10 Watts, 363; Bac. Abr. Limitation of Actions, D 3.
Generally, when the taxpayer receives income, it must accrue that amount, unless the receipt is subject to substantial limits or restrictions or is a deposit or loan.
For federal income tax purposes the California franchise tax continues to accrue in the California tax year.
461-1(a)(2), the taxpayer should properly accrue and deduct the gross receipts tax imposed by the State of Maryland in the calendar year following the end of the income year.
It was this penalty interest that the taxpayer sought to accrue, and which the IRS disallowed.
1991, a taxpayer was held to the contract terms of a loan document and was required to accrue interest when the taxpayer did not formally declare the notes to be in default.
Interest accrues between the date of the overpayment and the date the tax is credited or refunded (with a 30-day window for refund checks).
Example 2: XYZ, an accrual-basis partnership, accrues an otherwise deductible expense to A Corporation, a cash-basis C corporation.
Some of these arrangements also provide for "interest" to accrue on the amounts deferred until the benefits are paid to the participants.
1272 provides for the current inclusion of OID in income as it accrues regardless of a holder's method of accounting.


text:
No Sign of Retail's Death in Northern Colorado – by Jake Hallauer
NAI AffinityAugust 1, 2018November 18th, 2021
Despite what sensational headlines and reports from numerous media sources might suggest, retail is far from dead in Northern Colorado. In fact, the retail market is impressively vibrant. There is certainly some truth to the doom and gloom headlines relative to Northern Colorado, but generally those headlines are more applicable to certain segments of the retail market, such as anchor and junior anchor expansion, than the overall market.
With that said, there have been a number of significant anchor and junior anchor transactions over the past few years with SCHEELS opening their new 250,000 square foot megastore at Johnstown Plaza (which also includes Burlington, Ulta Beauty and Tuesday Morning), within the 2534 development at the southeast corner of I-25 and US Highway 34 just under a year ago, Lucky's Market under construction on their build-out at the former Sports Authority box at the northwest corner of College Avenue and Mulberry Street in Fort Collins, 24 Hour Fitness recently leasing the former Safeway box just east across College Avenue from Lucky's, Dick's Sporting Goods set to open later this month at the Foothills Mall redevelopment in Fort Collins and Ulta Beauty and 2nd & Charles recently leasing space at Front Range Village in Fort Collins. Some of those boxes sat vacant for years, however, and there are still a number of anchor and junior anchor spaces vacant or which will likely come vacant in the Northern Colorado market in the not so distant future.
Absorption of large new, or redeveloped, retail centers has been relatively strong over the past few years with the vast majority of new space being leased, despite a relatively high level of new construction with both the Foothills Mall redevelopment and Johnstown Plaza in lease-up. This has led to a very low retail vacancy rate in Northern Colorado – approximately 3.8% in at the end of Q2, 2018 according to CoStar. That rate declined from approximately 5.0% to start 2015 while Foothills and Johnstown Plaza, as well as many smaller retail developments have been in lease-up. With a vacancy rate well under 5.0%, nearly all of the new space being built absorbing, what appears to be significant demand from smaller tenants and decent demand from larger tenants and continued population and employment growth, "the death of retail" is more of a myth than a reality in Northern Colorado.
Retail is evolving in Northern Colorado, rather than dying. Investors and developers are looking at opportunities for mixed-use centers which include housing, entertainment, hotels and other uses which may function as "anchors" rather than the traditional retail anchors. This is evident with the 405 unit Cycle apartments McWhinney is developing adjacent to Foothills Mall, several apartment communities which exist or which are under construction adjacent to Johnstown Plaza and the Brands at the Ranch project in Loveland which is under construction with plans for a theater, apartments, hotels and an arena in additional to retail and restaurants. I anticipate that as this evolution occurs, there will likely be significant additional demand for regional distribution and warehouse space as retailers continue to "right size" their stores and shift their strategies to attempt to compete more favorably with Amazon.
With a rapidly growing population, which is projected to nearly double by 2035, a very low vacancy rate and decent demand for additional retail and restaurant space, I anticipate that Northern Colorado's retail market will continue to outperform many other markets throughout the country and "the death of retail" will remain a myth in the region for the foreseeable future. I expect the retail real estate evolution to continue and walk-able, mixed-use, community and neighborhood centers will likely be where much of the future retail is developed in the future.
Author: Jake Hallauer


text:
Q: Safari and IE ajax issues with subsequent calls (.then) I've got this code:
var request1 = this.add(prod_id, pack_id_combination, button, pack_qty).then(function(rdata1){
  // this always works
  this.add(prod_id, case_id_combination, button, case_qty).then(function(rdata2){
    // this sometimes (random) doesn't work
  }.bind(this), function(req, status, error){ console.log('case error',req, status, error); });
}.bind(this), function(){ console.log('pack error'); });

I omitted the rest of the code because it's a huge class and it's also  impossible to build a jsfiddle, too much things.
But here's the facts:

*

*my js points to a php file returning data. It does ajax calls via the add() function (see below)

*it always works for the first call, it sometimes doesn't work for the second call.

*on the log, status = 'error', error = '' (empty, so no info here)

*I also know for sure that the second add(), if called alone with the same params, will always work.

*Furthermore, it has these problems ONLY in safari and IE, not in chrome, ff, android

*Furthermore, it has these problems ONLY if tested online, while in my local environment always works properly.

*Safari says it looses the connection. This is the only information I can get
the add() function is quite trivial:
add:function(idProduct, idCombination, callerElement, quantity){
    return $.ajax({
        type: 'POST'
        ,headers: { "cache-control": "no-cache" }
        ,url: baseUri + '?rand=' + new Date().getTime()
        ,cache: false
        ,dataType : "json"
        ,data: 'controller=cart&add=1&ajax=true&qty=' + ((quantity && quantity != null) ? quantity : '1') + '&id_product=' + idProduct + '&token=' + static_token + ( (parseInt(idCombination) && idCombination != null) ? '&ipa=' + parseInt(idCombination): '')
        ,success:function(data, status, req){
            console.log('ok');
        }
        ,error:function(jqXHR, textStatus, errorThrown){
            console.log(jqXHR.responseText, textStatus, errorThrown);
        }
    });
},//end add()

So, my question: is there a way to avoid this problem? Is it a common pitfall? Are there common practices to avoid this kind of issues?

A: Ok, my issue was solved by using a GET request instead of a POST one.
Probably both Safari and IE are not able to correctly manage 2 subsequent ajax POST requests, at least in specific circumstances. Or, the jquery 2.1.3 (version used here) implementation has some kind of subtle  problems/bugs.
But since I often have random css/js issues with both IE (surprise...) and Safari (yes...), I guess it's them and not jquery.

redpajama_set_name: RedPajamaStackExchange


text:
Q: Is there a elegant way to convert a positive number to a +1 and a negative number to a -1? I have a high-performance component that needs to convert a values to -1, +1 or 0.
    static int ToOne(double value) {
        return (int)(value == 0D ? value : value / Math.Abs(value));
    }

The above methods are not elegant, is there any other way to do this?

A: You have basically recreated Math.Sign:

Returns an integer that indicates the sign of a number.

Return value    Meaning
-1              value is less than zero.
0               value is equal to zero.
1               value is greater than zero.

Test code:
Console.WriteLine(ToOne(-5));
Console.WriteLine(ToOne(-0.1));
Console.WriteLine(ToOne(0));
Console.WriteLine(ToOne(0.1));
Console.WriteLine(ToOne(5));

Outputs:
-1
-1
0
1
1

And with Math.Sign:
Console.WriteLine(Math.Sign(-5));
Console.WriteLine(Math.Sign(-0.1));
Console.WriteLine(Math.Sign(0));
Console.WriteLine(Math.Sign(0.1));
Console.WriteLine(Math.Sign(5));

Outputs:
-1
-1
0
1
1


Licensing for water restoration is not required in MARYLAND, MD. Water restoration certification can be a requirement to perform remediation projects for banking, property management, government. Water restoration training and certification shows the industry that you qualify to take on projects. For more detailed information about enrolling in a water restoration class or if you want more detailed information on the topics and what the training will include select the link to visit our Water Damage Class Information Page.


In reviewing several recent posts from non-healthcare publications, I began to consider, what should we expect in the content marketing world for healthcare marketing for 2019?
The world of content marketing is rapidly changing. As healthcare marketers, it is important that we have focus and consider the trends that other businesses will also experience.
First, a few general comments… content marketing is not going anywhere – this industry will soon be worth more than $400 billion. Content must complement your business strategy and strategic plan in achieving specific goals rather than a universal or general brand message. Good storytelling is at the core of all content strategy. Distribution is the driver of success—target the right people on the right channels. And we must continue to repurpose our best content.
Optimize SEO at all times – this determines the ranking of your content and website within search engines and it remains critical no matter what.
Continue to prioritize and improve your mobile strategy.
Be sure your website is easy to navigate and reflects your current brand.
Video, Video, Video – of course video is a digital tool, but it is so important it must be called out. This trend continues to be paramount in your content marketing strategy. The Washington Post predicted that video would account for 80% of online content by 2020. YouTube is already overtaking Facebook as the second-most visited site (Google is number one, naturally).
A note about podcasts – if you aren't doing them, you should consider it. This is an effective way to get your message out. This works very well as a recruitment and retention strategy. Consider interviewing employees to carry the message about working at your organization. This is especially helpful in nurse recruitment and/or in areas where there are critical shortages. And if you are doing podcasts, amplify them with embedded video.
Social Media Networks – DUH! We all understand how social media networks have taken over every aspect of what we do today. It naturally must be one of the core foundations of your content marketing strategy. One area that you may not have considered though is the use of social network community groups. Depending on your organization's strategic plan you can create groups. To learn more about how you can effectively implement at your healthcare organization see, Dee Dee Becker's blog post from November 13, 2018.
Influencer and Micro-Influencer Marketing – this will continue to grow. In industries outside of healthcare this has grown significantly on Instagram, Facebook and YouTube. This practice is still in its infancy. Some large healthcare organizations have taken advantage of this strategy early – such as Toby Cosgrove and the Cleveland Clinic Foundation – and have done so very effectively. This can also be done by key targeted smaller populations such as in the children's hospital space by moms and/or doctor moms. Additionally, you should consider one of your best and largest populations: your employees.
Internal Communications Strategy and Content – be sure that your internal communications strategy includes an employee "communications ambassador" program. Employees are your best source as storytellers. Strategize how to activate your employees as both creators and as effective distribution channels. Referring back to the trend of micro-influencer marketing, when your employees are engaged they will themselves act as micro-influencers for your company. If you are successful in employing and activating individuals who align with your brand values, they will act as your most dedicated cheerleaders. This creates special challenges for healthcare organizations and must be handled with great care – but it can be done very effectively.
Educate Yourself and the C-Suite about Advanced AI.
Identify best practices in healthcare for voice powered EVERYTHING!
Understand and use chatbots for the patient journey and population health strategies. These will become a customer service standard and will start replacing live agents more and more frequently as machine learning algorithms grow more complex and are able to emulate human beings with almost spooky accuracy.
I hope you find this information useful in planning for 2019 at your organization. What you do think will be the biggest trends in healthcare content marketing? Your questions and comments are encouraged and appreciated. Drop me an email at cindy@corriganconsulting.com.
Michael Brenner, "My 2019 Content Prediction," Marketing Insider, Content Marketing November 12, 2018.
Harry Southworth, "10 biggest Content Marketing Trends for 2019," Analysis, Public Relations, November 19, 2018.
Dee Dee Becker, "Leveraging Facebook Groups for your Health System, Hospital or Medical Group," Corrigan Consulting Blog, November 13, 2018.


This week's topic is to create a top 10 of the books you read this year!
Pivot Point, Looking for Alaska, and Reboot were some of my favorites of the year as well! Such awesome books.
Yes!!!! I loved them all!
I actually have The Unbecoming of Mara Dyer on my bookshelf but haven't read it yet but I loved Pivot Point!! Thanks for stopping by my blog and have great New Years!
You should totally read it! It's amazing!
Hopefully you can get to it! It's really good!
Great list. You've got some really amazing books on your list – I loved Anna and the French Kiss. I really want to read Pivot Point and Angelfall, was really happy to get both of those for Christmas. Happy New Year 🙂 My TTT.
Thanks! Yay! Hope you can get to them soon!
I just picked up Pivot Point and I can't wait to read it! Great list!


text:
Lahore, Pakistan in Mourning as many more die as a result of Easter Blast
begin quote from:
Pakistan in Mourning as Toll From Lahore Suicide Bombing Rises
LAHORE — Shock and grief enveloped Pakistan on Monday as the official death toll from a suicide attack in Lahore a day earlier rose to 69, with 341 people wounded. The local news media put the number of people killed at 71. …
By DANIYAL HASSAN and SALMAN MASOOD MARCH 28, 2016
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Family members mourned on Monday in Lahore, Pakistan, after a suicide bombing in a park a day earlier killed dozens. Credit Arif Ali/Agence France-Presse — Getty Images
LAHORE, Pakistan — Shock and grief enveloped Pakistan on Monday as the official death toll from a suicide attack in Lahore a day earlier rose to 69, with 341 people wounded.
The local news media put the number of people killed at 71.
Police investigators said a suicide bomber had detonated explosives in a vest during the evening rush hour on Sunday at Gulshan-e-Iqbal Park, one of the largest public parks in this eastern city.
Jamaat-e-Ahrar, a splinter group of the Pakistani Taliban, claimed responsibility for the attack, saying that it had targeted Christians. Pakistani officials were skeptical about the claim, as most of those killed and wounded were Muslims. But because Sunday was Easter, a large number of Christian families had come to the park.
Years of Terror in Pakistan
The suicide bombing in Lahore, Pakistan, on Sunday is only one of dozens of attacks that have killed hundreds of civilians in Pakistan in recent years, with the Taliban or its splinter groups claiming responsibility for many.
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Prime Minister Nawaz Sharif arrived in Lahore on Monday morning and visited Jinnah Hospital to show solidarity with the victims. He said he was deeply grieved and vowed to bring the culprits to justice.
Later, he met with senior government officials and pledged to eliminate terrorism.
"Our resolve as a nation and as a government is getting stronger, and the cowardly enemy is trying for soft targets," Mr. Sharif said during the meeting. "Our goal is not only to eliminate terror infrastructure but also the extremist mind-set, which is a threat to our way of life."
Mr. Sharif was planning to address the nation Monday evening, his press office said.
The military's chief spokesman, Lt. Gen Asim Saleem Bajwa, said Monday that intelligence raids had been carried out in three cities in Punjab Province, of which Lahore is the capital, after the bomb attack, and a "number of terrorists and facilitators were arrested." However, he gave no further details about the identities of those arrested or whether they were connected to Sunday's bombing.
According to hospital sources, 252 people had been admitted to hospitals in the city by 10 a.m. on Monday, while 89 had been discharged. Of the wounded, 26 were in critical condition, officials said.
where bomb
went off
Gulshan-e-Iqbal
of the Punjab
Gulshan-e-Iqbal Park
An initial police report said that four young men had been approached by a police officer near the entrance to the park on Sunday evening. Three of them managed to escape, but the fourth ran through the gate and detonated his explosives. At least 10 kilograms of explosives was used in the bombing, the police said.
Lahore, widely considered the cultural and political capital of Pakistan, appeared in mourning on Monday. Most of the commercial centers and shopping areas were closed, and security forces were on high alert throughout the city. A three-day Spring festival at Race Course Park in Lahore was cancelled because of security fears.
The police cordoned off the blast site for forensic investigation. On Monday afternoon, shoes and shreds of clothing were strewn about in the parking lot of the park.
At city hospitals, chaos and anxiety prevailed as relatives kept pouring in to visit the wounded victims.
The site of the attack. Credit Rahat Dar/European Pressphoto Agency
Rashida Bibi, 50, who was being treated for head injuries, said she and 32 members of her extended family had come to Lahore from Sahiwal, another city in Punjab, to enjoy Sunday. "We were at the swings. Suddenly there was a blast and I fell down. Children and women started screaming and soon rescue workers arrived. I cannot describe the terrible scenes," Ms. Bibi said.
Meanwhile, at least 2,000 protesters continued a sit-in outside the Parliament in the national capital, Islamabad.
The protesters had rampaged through the city on Sunday after holding a rally in neighboring Rawalpindi to express support for Malik Mumtaz Hussain Qadri, a former police guard who was executed on Feb. 29 for the 2011 killing of a governor, Salmaan Taseer. Mr. Taseer had called for changes in the country's blasphemy laws, saying they were being used to persecute religious minorities.
To hard-line Islamists and religious parties, any change to the blasphemy laws is unacceptable, and they have campaigned violently against such proposals, portraying Mr. Qadri as a hero
News Clips By REUTERS and THE ASSOCIATED PRESS 00:47
Aftermath of Pakistan Bombing
Dozens of people, including children, were killed and at least 300 were wounded in an apparent suicide blast in a park in Lahore, Pakistan.
By REUTERS and THE ASSOCIATED PRESS on Publish Date March 27, 2016. Photo by Rahat Dar/European Pressphoto Agency. Watch in Times Video »
On Monday, the protesters presented their demands to the government. They included declaring Mr. Qadri an official martyr, imposition of Sharia law in Pakistan and immediate execution of all those convicted of blasphemy.
District officials in Islamabad and leaders of the protesters were in negotiationson Monday afternoon to end the sit-in. However, political analysts said they expected a protracted standoff, as the government was unlikely to accede to the demands.
Troops had been deployed in Islamabad to secure the Parliament and other important buildings, including the headquarters of the Supreme Court and a large complex of apartment blocks for Parliament members.
Mobile phone service in some parts of the capital was suspended by the authorities.
The attack in Lahore, which is also Mr. Sahrif's political stronghold, has drawn new attention to the government's efforts to stem terror in the country and has renewed calls for action against militant groups in Punjab.
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Previously, Mr. Sharif and his younger brother, Shahbaz Sharif, the chief minister of Punjab, had resisted calls for an army operation in the province.
Senior police officials had also refused to consider military action, saying that the provincial police are capable of handling the militants.
Critics have faulted the government's handling of protests in the wake of Mr. Qadri's execution. Omar R. Quraishi, a senior journalist based in the port city of Karachi, said the government's policy of letting pro-Qadri supporters vent their fury seemed to have backfired.
"Calling in the army to re-establish peace in Islamabad is a sign of the government's inability to govern effectively," Mr. Quraishi said.
Follow Salman Masood on Twitter @salmanmasood.
Daniyal Hassan reported from Lahore, Pakistan, and Salman Masood from Islamabad, Pakistan.
Posted by intuitivefred888 at 10:06 AM
Labels: Lahore, Pakistan in Mourning as many more die as a result of Easter Blast
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Libby becomes Candace's eighth-grade teacher. The school, the town, their lives are difficult for both of them and, in finding each other, they share an understanding.
But a student-teacher relationship is never simple: are they actually good or bad for each other?
The Orchestras offers a meditative, reflective example of the educational system and life in a small Southern town, taken from Aaron C. Wong's teaching experiences in Belzoni, Mississippi.


text:
Q: Reduction in execution time when under-subscribing on Google Cloud I have an executable that supports multithreading and I'm trying to execute it on Google Cloud. I've reserved 8 VCPUs and I'm executing the job using 8 threads. Let's say that I get an execution time of y. Now I've reserved 16 VCPUs, but am only executing the job using 8 threads, and I get an execution time of x. What I'm noticing is that x is almost 15-20% less than y. Why do I get this performance benefit when I reserve more VCPUs, but use less threads?
Any help will be appreciated. Thanks.

A: At Google Cloud Platform, The performance of one virtual CPU is not equal to the one physical core. More preciously, One virtual CPU == 1 hyper-thread == 1/2 physical core.
With hyper-threading, one physical core appears as two processors to the operating system, allowing concurrent scheduling of two processes per core.
So when you used 16 vCPUs meaning you have 16 hyper-thread and 8 physical core whereas 8 vCPUs supported by 4 physical core and have 8 hyper-thread. So it means you have more processing capacity when using 16 vCPUs in regards to 8 vCPU which also reflects the performance benefit you are getting.
This documentation in StackOverflow, Blog and Google Group discussions are a good read on similar topic.


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Originally Published: June 9, 2018 6:01 a.m.
Two young men have been arrested in Prescott Valley after the suspected drug overdose of 18-year-old Jared Friederick, according to Prescott Valley Police Department (PVPD) officials.
Friederick was found not breathing and without a pulse on the floor of a residence in the 8400 block of East Leigh Drive of Prescott Valley by PVPD officers shortly before 5 p.m. Thursday, June 7, according to a news release.
The officers immediately conducted appropriate first aid procedures until a Central Arizona Fire and Medical Authority crew arrived and took over medical treatment. Friederick was transported to the Yavapai Regional Medical Center campus in Prescott Valley, where he was pronounced dead, police said.
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As a result of the investigation, Rick Alexander Arellano, 18, and Lester James Suter, 18, were arrested and charged with multiple charges including possession of narcotics for sale and possession of drug paraphernalia, both felonies. Both men were booked into the Yavapai County jail. The investigation is continuing.
"If you're aware of any possible violations of drug use, you are encouraged to immediately contact law enforcement," said PVPD spokesperson Jerry Ferguson.


text:
Virginia Conservation Police Notebook
To increase awareness of Conservation Police Officers (CPO's-previously called game wardens) activities, the "Virginia Conservation Police Notebook" provides an overview of activities encountered by our officers who protect natural resources and people pursuing outdoor recreation in the fields, woods and waters of Virginia. These reports are prepared from the officer's field notes by Kim McCarthy, Executive Assistant to Major Scott Naff [Operations] and Major Bryan Young [Administration] of the Law Enforcement Division of DGIF. These CPO reports show the value of concerned citizens, landowners and true sportsmen in providing tips to law enforcement officers on suspected violations by lawbreakers who give other outdoor enthusiasts an undeserved bad reputation.
Region I – Tidewater
Trout Fishing Debacles – On January 18, 2020 CPO Brandon Robinson received a call for service from an anonymous caller regarding an angler who frequently fishes Shield's Lake in Richmond, which is part of the urban trout fishery. In the call for service, there was a link to the angler's Youtube channel. While watching his Youtube videos, CPO Robinson was able to detect several violations. The first video was dated 1/14/20. The video showed the angler netting a 21" rainbow trout swimming near the surface of the water. In the video, the angler stated that he knew it was illegal to catch the fish in the net, but he wanted to eat it. While watching another video dated 12/22/19, the fishermen foul hooked a brown trout and failed to release the fish back into the lake, and again stated that he knew it was illegal to creel this fish. It was also determined the angler did not purchase a trout license until 1/09/20. While speaking to the fishermen, he admitted to fishing at Shields Lake several times before purchasing a trout license. In the same video, recorded on 12/22/19, the fishermen told his viewers that "not only did they need a freshwater fishing license, but they also needed a trout license" to fish at the lake, even though at the time he did not possess a valid trout license himself. Appropriate charges were placed.
Waterfowl Hunter Caught Killing Seagull – On January 26, 2020, CPO Chambers was on patrol at Back Creek Boat Ramp in York County when he heard shooting coming from up the creek. From the end of the pier, he observed a decoy spread and two hunters standing on the shore in a small blind made from brush. Officer Chambers observed the hunters for about an hour making notes of their actions. The two hunters shot at a number of ducks and geese. Shortly after one of the hunters' weapons jammed, the other hunter shot and killed a passing seagull. The hunter with the jammed weapon paddled out to the seagull in a kayak, retrieved the bird, and hid it behind the bank away from their blind. CPO Chambers made contact with the hunters who initially lied about hunting and shooting the seagull. After informing them they had been under observation for some time, they became cooperative. The hunter who shot the seagull did not have any hunting licenses and also had an unplugged shotgun. The other hunter did not have his hunting license and did not have a PFD for the kayak they were using to get to the blind. Both hunters were trespassing as well. Multiple charges were placed for the license and hunting violations.
Waterfowl Hunters Charged with Numerous Violations after the Season Ends – On January 31, 2020, CPO Brian Bratton and Sergeant Garvis launched their 19' patrol boat for one last evening waterfowl patrol of the 2019-2020 season. With 20 minutes remaining in the season they observed a boat come from the boat ramp and head up into the large coastal bay marshes of Jobes Island. From the close of the season, at 1725 hours, the officers observed the boat occupants attempt to take migratory waterfowl after legal shooting hours, shooting under power of the outboard motor and trespassing to hunt. At this point, the officers set up to intercept the hunters as they returned to the boat ramp. They observed the boat approach their position well after sunset operating without the required navigation lights. After stopping the boat the officers found that the occupants were in possession of lead shot and one hunter had a shotgun capable of holding more than 3 shells. CPO Bratton was well acquainted with the two individuals as he had apprehended both of them in the same area for placing and hunting migratory waterfowl with the aid of bait and trespassing in November. Federal charges are pending, most likely to be combined with the already pending case from November.
CPO Catches Poachers Shooting Snow Geese from the Road – On January 31, 2020, CPO Machen received information that an individual had shot into a flock of snow geese feeding close to Seaside Rd. in Northampton County. The caller provided a picture of the vehicle that included the tag number. Using this information, CPOs Machen and Pennino developed and located the suspect. CPO Pennino was familiar with the individual as well as with other family members as they were involved in hunting violations earlier in the season. During the field interview the suspect admitted to running out into the field, while carrying his shotgun to retrieve a snow goose. However, he would not admit to shooting the goose. Based on evidence collected, charges are pending for trespassing to hunt and shooting from the road.
Snow Geese Shot from Moving Vehicle – The next afternoon CPO Machen was on patrol in the same area and observed a flock of approximately 3,000 snow geese again feeding in the wheat field very close to the Seaside Rd. CPO Machen took up a concealed observation post and waited. After approximately 1 hour, Machen observed a pickup truck traveling north on the Seaside Rd. As the truck approached the field where the geese were feeding, he observed the passenger window of the truck go down, a barrel of a firearm come out and heard several gun shots. CPO Machen noted this all occurred while the tuck was still traveling approximately 30 to 40 mph. Due to the speed the vehicle was travelling and the distance CPO Machen had to travel to get to the road, Officer Machen was unable to stop the suspects before they left the immediate area. At this point, CPO Machen returned to his hiding spot convinced the suspects would return. They did and CPO Machen was able to get their vehicle stopped. During the field interview both suspects confessed and admitted that they had made a bad mistake by shooting the snow geese from the vehicle. The appropriate charges were placed.
Region II – Southside
Snagging and Littering – CPOs John Daniel and Andrew Howald worked a special operation on a trout stream in Amherst County after a stocking in late January. They observed an angler, whom they were familiar with, snag a good-sized trout, and upon reeling it in, claim that he had foul hooked it, did not mean to, and was not going to keep it; however, he then turned his back to other anglers and put the trout in his creel. Before the officers made contact with him, they also observed his fishing partner litter in the stream. Both men were approached and charges were placed for snagging and littering the public waterways.
CPO's Assist with Tracking of Suicidal Subject – Campbell County Sheriff's Office requested DGIF K9 assistance tracking a suicidal subject who had cut himself, thereby leaving a blood trail in the woods. K9 CPO Richard Howald responded with his partner Skyy long with CPO's John Daniel and Stephen Ritchie. The officers began tracking with two Campbell deputies and a Bedford deputy who had his patrol K9. Shortly into the track, the subject came walking out of the brush, after hearing the dogs, and stated that he was going back home and wasn't going with anyone. The Campbell deputies approached him as the CPO's and Bedford deputy were giving him commands. CPO Howald had to put him in a control hold to keep him from striking the deputy and then was able to get him handcuffed. The subject was taken into custody by the deputies and treated by EMS. No officers were injured in the event.
Region IV – Mountains & Shenandoah Valley and Northern Piedmont
Charges Pending on Multiple Violations – On the evening of Friday, January 31, 2020, Sgt. Rich Goszka and Sr. Officer Mark Sanitra were off duty when they received a call from dispatch in regards to a person placing and hunting over bait in Prince William County. The officers were able to apprehend the suspect leaving the area by vehicle. The suspect confessed to placing the bait and to killing a buck during archery season. The officers followed the suspect to his residence and the buck was seized. Violations discovered were placing bait, hunting without blaze orange, hunting without licenses, failing to check a deer and a CWD importation violation. Charges are pending.
Team Effort Catches Poacher – Districts 41 and 47 worked together to investigate a subject suspected of a multitude of game law violations. Through a variety of investigative techniques, including a search warrant, CPOs collected evidence related to such offenses as trespassing to hunt, exceeding the season limit for deer, hunting during the closed season, etc. This individual has had a detrimental impact on the wildlife resource in multiple counties, and is well known for violations in multiple districts. This lengthy, on-going investigation will serve to conserve and protect our wildlife in the future.
Cane Toad Licking – On Saturday, February 2, 2020, Master CPO Ray Solomon responded to information provided by the Prince William County Animal Control in regards to the illegal possession of exotic wildlife. The investigation revealed that the Prince William County Police responded to a disturbance in a neighborhood. When the police arrived a 20 year old male was acting disorderly. The police had the male transported to the hospital as he was acting as if he was under the influence of a drug. The man was very incoherent and combative even breaking the restraints. Police discovered that he recently purchased two cane toads and may have licked them to get high. Bufotenin, one of the chemicals excreted by the cane toad, is classified as a controlled substance. The effects of bufotenin are thought to be similar to those of mild poisoning; the stimulation, which includes mild hallucinations, lasts less than an hour. As the cane toad excretes bufotenin in small amounts, and other toxins in relatively large quantities, toad licking could result in serious illness or death. Solomon interviewed the suspect's father who voluntarily surrendered the two cane toads after he was informed they were illegal to possess in Virginia. The toads were purchased online and shipped from Florida. A charge is pending.
Great Horned Owl Recovered – On February 4th, 2020, DGIF was contacted by the Prince William Animal Control regarding an owl that was observed in a dog kennel in the backyard of a residence. Based on information provided Officer Eric Plaster, a search warrant was obtained for the property. On February 5th Officer Plaster, Sr. Officer Sanitra and biologist Joran Green executed the search warrant. A great horned owl was recovered from a dog pen. The owl was transported to a wildlife rehabilitation center for evaluation. The investigation is ongoing.


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text:
\section{introduction}

\noindent It is a well-known fact that for a countable group, the homological dimension and the  cohomological dimension over a non-zero commutative ring $R$ differ by at most one, and that in general the homological dimension is always less than or equal to the cohomological dimension, see, for example \cite[Theorem 4.6]{bieribook}. Emmanouil and Talelli \cite[Theorem 2.1]{emmanouil-12} give a criterion  for a group of finite integral homological dimension to have both quantities  equal. The main ingredient for this result is the following:

\begin{theorem}\label{main-et}\cite[Theorem 1.3]{emmanouil-12}
Let $R$ be a countable ring and $M$ be a countably generated flat left $R$-module. Then the following conditions are equivalent:
\begin{enumerate}
\item $M$ is projective.
\item $\Ext_R^1(M,R)=0$.
\item $\Ext_R^1(M,R)$ is a countable group.
\end{enumerate}
\end{theorem}

In this paper we will verify a  generalisation of this theorem to the setting of Bredon-cohomology. Here the group $G$ is replaced with the orbit category ${\OC{G}{\frakF}}$, whose
objects are the transitive $G$-sets $G/H$ with $H\in \mathfrak{F}$ and whose
morphisms are $G$-maps. The category $\mathop{\calO_{\frakF}G\text{-}{\Mod}}$ of right Bredon modules is the
functor category whose objects are contravariant functors $M\: {\OC{G}{\frakF}}\to \Ab$
from the orbit category to the category $\Ab$ of abelian groups. As a functor
category  it is, for example,  an abelian category which satisfies the same
Grothendieck Axioms as the category $\Ab,$ and which has enough projective and
enough injective objects. Hence there is a natural setting to do homological algebra.
In the next section we briefly introduce those notions from Bredon-cohomology needed here. We refer the reader to~\cite{luck-89, mislin-03} and also
to the PhD thesis of the first author~\cite{fluch-phdthesis} for more detail. We prove:

\begin{Theorem A}
Let $G$ be a countable group, $\mathfrak{F}$ a family of subgroups of $G$ of finite
conjugacy type and let~$\mathcal{F}$ be a complete set of representatives of the
conjugacy classes of the elements of~$\mathfrak{F}$. If $n := \hd_\mathfrak{F} G$ is
finite, then the following statements are equivalent:

\begin{enumerate}
\item $\cd_\mathfrak{F} G = n $

\medskip

\item $H^{n+1}_\mathfrak{F} \Bigl(G; \prod\limits_{H\in \mathcal{F}} \mathbb{Z}[-,
G/H]_G\Bigr) = 0$

\item $H^{n+1}_\mathfrak{F} \Bigl(G; \prod\limits_{H\in \mathcal{F}} \mathbb{Z}[-,
G/H]_G\Bigr)$ is countable.
\end{enumerate}
\end{Theorem A}

Note that for $\mathfrak{F}=\{1\}$ this is
\cite[Theorem 2.1]{emmanouil-12}.

The main ingredients needed in the proof are the facts that a flat contravariant $\OC{G}{\frakF}$-module is a colimit of finitely generated free $\OC{G}{\frakF}$-modules \cite[Theorem~3.2]{nucinkis-04}, and that certain limits of duals of free $\OC{G}{\frakF}$-modules satisfy the Mittag--Leffler condition for Bredon-modules. This is defined analogously to the case of modules over a commutative ring, and we shall give precise definitions and properties in Section \ref{ML-section}.

We will apply Theorem A to groups of Bredon-homological dimension $1$. It is a well-known conjecture, attributed to R.~Bieri, that a finitely generated group of integral homological dimension $1$ is free. In \cite{bridson-12} Bridson and Kropholler remark that  a naive generalisation to rational homological dimension is not true as the case of the lamplighter group shows. In Section \ref{Bredonhom-1} we ask a related  question for Bredon homology, where we replace the condition that the group is finitely generated by the property Bredon-$\FP_1$. We shall answer the question in some specific situations.

Finally, we  apply Theorem \ref{main-et} to the rational cohomology of the group and will show a virtual version of \cite[Proposition 3.2]{emmanouil-12}.

\subsection*{Acknowledgements}

The authors would like to thank the referee for many helpful comments and for spotting an error in an earlier version. The first author gratefully acknowledges the support through the SFB~701 in Bielefeld. The second author acknowledges the support by the Danish National Research Foundation (DNFS) through the Centre for Symmetry and Deformation.

\section{Preliminaries on Bredon cohomology}

Let $G$ be a group and let $\mathfrak{F}$ be a family of subgroups closed under taking subgroups and under conjugation. ${\OC{G}{\frakF}}$ denotes the  category with elements the transitive $G$-sets with stabilisers in $\mathfrak{F}$ and morphisms, $\mor_{\mathfrak{F}}(x,y,)$ given by the set of $G$-maps between them.
We now form the free abelian group on the set of morphisms, which is denoted as follows:
$$\mathbb{Z}\mor_{\mathfrak{F}}(x,y) =\mathbb{Z}[x,y]_{\mathfrak{F}}.$$

Let $R$ denote a commutative ring with $1$.
The category of covariant ${\OC{G}{\frakF}}$-modules, denoted $\mathop{{\Mod_R\text{-}}{\OFG}}$, is now defined to be the category of covariant additive functors from ${\OC{G}{\frakF}}$ to the category of left $R$-modules. Analogously we define the category of contravariant ${\OC{G}{\frakF}}$-modules, denoted ${{\OFG}\text{-}{\Mod_R}}$. If the variance is clear from the context, or a statement is valid for either category, we simply talk of ${\OC{G}{\frakF}}$-modules.

The category of ${\OC{G}{\frakF}}$-modules inherits all of Grothendieck's axioms for an abelian category that are satisfied by the category of $R$-modules. Short exact sequences are evaluated point-wise, so are limits and colimits. We have the usual categorical tensor-product: let $M\in \mathop{{\Mod_R\text{-}}{\OFG}}$ and $N \in {{\OFG}\text{-}{\Mod_R}}$, then the tensor product is denoted by
$$M \otimes_{\mathfrak{F}} N.$$

We can form contravariant and covariant ${\OC{G}{\frakF}}$-modules $R[-,y]_{\mathfrak{F}}$ and $R[x,-]_{\mathfrak{F}}$ respectively:
$$R[-,y]_{\mathfrak{F}}(x) = R\otimes_\mathbb{Z} \mathbb{Z}[x,y]_{\mathfrak{F}}\phantom{.}$$
and
$$R[x,-]_{\mathfrak{F}}(y) = R\otimes_\mathbb{Z} \mathbb{Z}[x,y]_{\mathfrak{F}}.$$

\subsection{Free ${\OC{G}{\frakF}}$-modules} We have the usual notion of free, projective and flat modules. In particular, the modules $R[-,y]_{\mathfrak{F}}$ and $R[x,-]_{\mathfrak{F}}$ are free, and any free is a direct sum of modules of this form. For detail the reader is referred to \cite{luck-89} for the general set-up, and to \cite{fluch-phdthesis} and \cite{nucinkis-04} for this specific case.

Cohomology and Homology functors $\Ext_{\mathfrak{F}}^*(M,N)$ and $\Tor_*^{\mathfrak{F}}(M,N)$ are now defined in the usual manner.
We say a free $\OC{G}{\frakF}$-module $F$ is finitely generated, or countably generated, if there is a $G$-finite, respectively $G$-countable $G$-set $\Delta$ with finite stabilisers, such that~$F \cong \mathbb{Z}[-,\Delta]_\mathfrak{F}$.  For detail see \cite{luck-89, kropholler-09}.  We say an $\OC{G}{\frakF}$-module is finitely generated, countably generated if there is a finitely generated, respectively countably generated free module mapping onto it.

\subsection{Categories of finite type}

We say the category ${\OC{G}{\frakF}}$  is of \emph{finite type}, if there are finitely many objects $X=\{x_1,\ldots ,x_n\}$ such that for every object $y \in {\OC{G}{\frakF}}$ there exists $x_i \in X$ such that $\mor_{\OC{G}{\frakF}}(y,x_i) \neq 0$.
We say ${\OC{G}{\frakF}}$ is of \emph{finite isomorphism type} if there are finitely many objects $X=\{x_1,\ldots ,x_n\}$ and for each $y \in {\OC{G}{\frakF}}$ there is an isomorphism $\varphi \in \mor_{\OC{G}{\frakF}}(y,x_i)$ for some $x_i \in X$. This is equivalent to saying that $\OC{G}{\frakF}$ is of finite \emph{conjugacy type}, i.e.~that there are finitely many conjugacy classes of subgroups in $\mathfrak{F}$.

\begin{remark}
For $\mathfrak{F} = \frakF_{\fin}$ the family of finite subgroups, finite type and finite isomorphism type are the same and are equivalent to saying that the group is of type Bredon-$\FP_0$, i.e.~that the group has finitely many conjugacy classes of finite subgroups.

For $\mathfrak{F}=\frakF_{\vc}$ the family of virtually cyclic subgroups, on the other hand, Bredon-$\FP_0$ is equivalent to being of finite type \cite[Lemma 2.3]{desiconchabrita-11}, and it was shown in \cite{groveswilson} that an elementary amenable group of type Bredon-$\FP_0$ is necessarily virtually cyclic. Finite isomorphism type, on the other hand, is a much stronger condition. For example, the famous  of  construction Higmann--Neumann--Neumann \cite[6.4.6]{robinson}, is a group of finite type; in fact $\OC{G}{\frakF}$ has an initial object, but one can construct examples where the group is not of finite isomorphism type, see also \cite[Remark 2.1]{desiconchabrita-11}.

\end{remark}

\iffalse
\begin{lemma}\label{ctbl-pres}
Let $\OC{G}{\frakF}$ be a category of finite type. Then a  countably generated $\OC{G}{\frakF}$-module is  countably presented.
\end{lemma}

\begin{proof}

Observe  that  an argument analogous to that in \cite{kropholler-09}[Lemma 3.2]
\end{proof}

\fi

\subsection{Dualising ${\OC{G}{\frakF}}$-modules}
To generalise the usual notion of a dual module over a ring  to this
setting one needs to observe that ${\OC{G}{\frakF}}$-modules can be described as modules
over rings over several objects in the sense of~\cite{street-95}.

\iffalse{The
replacement of the group $G$ by the orbit category ${\OC{G}{\frakF}}$ translates into the
replacement of the group ring $\mathbb{Z} G$ by the Bredon bi-module $\mathbb{Z}[?,??]_G$,
which we call the \emph{Bredon ring} $\mathbb{Z}_\mathfrak{F} G$.}\fi

We replace the ring $R$ by a bi-functor
$$R{\OC{G}{\frakF}}\: {\OC{G}{\frakF}}^{\rm opp} \otimes {\OC{G}{\frakF}} \to \mathop{R\text{-}{\operator@font Mod}},$$
given by $R{\OC{G}{\frakF}}(x,y)=R[x,y]_{\mathfrak{F}}$. In \cite{street-95} this functor is denoted $\mathcal{H}_{\OC{G}{\frakF}}$.

\begin{DEF}
The \emph{dual module $M^*$} of a contravariant ${\OC{G}{\frakF}}$-module $M$ is defined as
\begin{equation*}
M^* := \Hom_{\OC{G}{\frakF}}(M, R{\OC{G}{\frakF}}),
\end{equation*}
which is a covariant ${\OC{G}{\frakF}}$-module. If $M$ is a covariant ${\OC{G}{\frakF}}$-module,
then its \emph{dual module~$M^*$} is defined in the same way, which yields a
contravariant ${\OC{G}{\frakF}}$-module.
\end{DEF}

In particular this  defines additive contravariant functors
\begin{align*}
(-)^*\: & {{\OFG}\text{-}{\Mod_R}} \to \mathop{{\Mod_R\text{-}}{\OFG}}
\\
\intertext{and}
(-)^*\: & \mathop{{\Mod_R\text{-}}{\OFG}} \to {{\OFG}\text{-}{\Mod_R}}
\end{align*}
which are exact. In particular injective morphisms are mapped to surjective
ones and vice versa.

\begin{observation}
Using Yoneda type arguments analogously to those in \cite{mislin-03, nucinkis-04} it follows directly that for
any $x\in {\OC{G}{\frakF}}$ the module dual to $M= R[x, -]_{\mathfrak{F}}$ is $M^* = R[-,x]_{\mathfrak{F}}$ and that the dual module of~$M = R[-, x]_{\mathfrak{F}}$ is $M^*= R[x,-]_{\mathfrak{F}}$.
\end{observation}

\begin{proposition}
\label{prop:dualising-bredon-modules}
Let $P$ be a finitely generated projective covariant ${\OC{G}{\frakF}}$-module.
\begin{enumerate}
\item $P^*$ is a finitely generated projective contravariant ${\OC{G}{\frakF}}$-module.

\item For any covariant ${\OC{G}{\frakF}}$-module $M$, there is a natural isomorphism
\begin{equation*}
\varphi\: P^*\otimes_{\OC{G}{\frakF}} M \to \Hom_{\OC{G}{\frakF}}(P,M)
\end{equation*}
of abelian groups.

\item For any contravariant ${\OC{G}{\frakF}}$-module $M$, there is a natural isomorphism
\begin{equation*}
\varphi'\: M\otimes_{\OC{G}{\frakF}} P \to \Hom_{\OC{G}{\frakF}}(P^*,M) \,.
\end{equation*}

\item There is a natural isomorphism
\begin{equation*}
 \varphi''\: P\to P^{**}.
\end{equation*}
\end{enumerate}
\end{proposition}

\begin{proof} The proof of the analogous statement for ordinary cohomology ~\cite[Proposition I.8.3.]{brown-82} works nearly verbatim
in the Bredon setting. This was checked for finitely generated free Bredon-modules in \cite{nucinkis-04}.  The result for projectives follows from the fact, that for finitely generated projective modules $P$, there is a finitely generated free $F$ and a module $Q$, such that $F \cong P\oplus Q$, and $F^* \cong P^* \oplus Q^*.$
\end{proof}

\subsection{Flat ${\OC{G}{\frakF}}$-modules} The following result was shown in  \cite[Theorem 3.2.]{nucinkis-04} for $\mathfrak{F} =\frakF_{\fin},$ but the proof works for arbitrary families $\mathfrak{F}.$

\begin{proposition}\label{lazard}
Let $M$ be a contravariant $\OC{G}{\frakF}$-module. Then the following are equivalent:
\begin{enumerate}
\item $M$ is flat.
\item $M$ is a colimit of finitely generated free $\OC{G}{\frakF}$-modules.
\end{enumerate}
\end{proposition}

\section{The Mittag--Leffler Condition}\label{ML-section}

The directed sets we consider in this article are all countable. Hence, when considering limits
or colimits of objects~$\{A_\alpha\}_{\alpha\in D}$ directed by a countable
directed set $D$, we may assume that
we are dealing with a \emph{tower}
\begin{equation*}
\ldots \to A_4 \to A_3 \to A_2 \to A_1 \to A_0
\end{equation*}
or respectively with a \emph{co-tower}
\begin{equation*}
\ldots \leftarrow A_4 \leftarrow A_3 \leftarrow A_2 \leftarrow A_1 \leftarrow
A_0
\end{equation*}
of objects.

\begin{DEF}
\cite[Definition 3.5.6.]{weibel-94}
\label{def:m-l}
Let $\{A_\alpha\}$ be a tower of objects in an abelian category.

\begin{enumerate}
\item The tower of objects is said to satisfy the \emph{Mittag--Leffler
condition,} if for any $k$ there exists a $j\geq k$ such that for any $i\geq j$
the image of $A_i\to A_k$ equals the image of $A_j\to A_k$.

\item The tower of objects is said to satisfy the \emph{strict (or trivial) Mittag--Leffler
condition,} if for any $k$ there exists a $j\geq k$ such that the map $A_j\to
A_k$ is zero.
\end{enumerate}
\end{DEF}

If the abelian category $\mathfrak{A}$  is cocomplete and has enough
injectives, then one can define the right derived functors of the
$\varprojlim$-functor. If in addition, the category $\mathfrak{A}$ satisfies the Grothendieck Axiom (AB4*) then
the following explicit construction of the right derived functors of
$\varprojlim$ works for \emph{countable} limits, that is limits of towers, see \cite[\S 3.5.]{weibel-94}:

\begin{enumerate}
\item $\varprojlim^0 A_\alpha := \varprojlim A_\alpha$

\item $\varprojlim^1 A_\alpha := \coker(\Delta)$ where $\Delta\:
\prod A_\alpha \to \prod A_\alpha$ is the homomorphism as defined
in~\cite[p.~81]{weibel-94}

\item $\varprojlim^n
A_\alpha := 0$ for $n\geq 2$.
\end{enumerate}

\begin{DEF}
\label{def:m-l-weak}
Consider the functor category $\mathfrak{A}^I$ where $\mathfrak{A}$ is an abelian category
and $I$ is a small category. Let $\{A_\alpha\}$ be a tower of objects in
$\mathfrak{A}^I$.

We say that $\{A_\alpha\}$ satisfies the (strict) Mittag--Leffler condition in
the \emph{weak sense,} if for every $i\in I$ the tower $\{A_\alpha(i)\}$ of
objects in $\mathfrak{A}$ statisfies the (strict) Mittag--Leffler condition.

If we want to emphasise that the tower $\{A_\alpha\}$ satisfies the (strict)
Mittag--Leffler condition not just in the weak sense, but as defined in
Definition~\ref{def:m-l}, then we may say that the (strict) Mittag--Leffler
condition is satisfied in the \emph{strong sense}.
\end{DEF}

\iffalse
\begin{example}
Let $G :=\mathbb{Z}^2$ and consider the family $\mathfrak{F} := \frakF_{\vc}(G)$ of virtually
cyclic subgroups of $G$. Let $\mathcal{M} := \{ H_0, H_1, H_2,\ldots\}$ be the
set of maximal cyclic subgroups of $G$. For each $n\in \mathbb{N}$ let
$\{A^{(n)}_\alpha\}$ be the tower given by
\begin{equation*}
A_\alpha^{(n)} :=
\begin{cases}
\mathbb{Z} & \text{if $n+1$ does not divide $\alpha$}
\\
0  & \text{if $n+1$ divides $\alpha$}
\end{cases}
\end{equation*}
and structure maps identities where ever possible. For example
for $n=0,1,2,3$ we obtain the towers
\newlength{\mylength}
\settowidth{\mylength}{$\mathbb{Z}$}
\newcommand{\makebox[\mylength]{$0$}}{\makebox[\mylength]{$0$}}
\begin{gather*}
\ldots \to \makebox[\mylength]{$0$} \to \makebox[\mylength]{$0$} \to \makebox[\mylength]{$0$} \to \makebox[\mylength]{$0$} \to \makebox[\mylength]{$0$} \to \makebox[\mylength]{$0$} \to \makebox[\mylength]{$0$} \to \makebox[\mylength]{$0$} \to \makebox[\mylength]{$0$}
\to 0
\\
\ldots \to \mathbb{Z} \to \makebox[\mylength]{$0$} \to \mathbb{Z} \to \makebox[\mylength]{$0$} \to \mathbb{Z} \to \makebox[\mylength]{$0$} \to \mathbb{Z} \to \makebox[\mylength]{$0$} \to \mathbb{Z} \to 0
\\
\ldots \to \makebox[\mylength]{$0$} \to \mathbb{Z} \to \mathbb{Z} \to \makebox[\mylength]{$0$} \to \mathbb{Z} \to \mathbb{Z} \to \makebox[\mylength]{$0$} \to \mathbb{Z} \to
\mathbb{Z} \to 0
\\
\ldots \to \mathbb{Z} \to \makebox[\mylength]{$0$} \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z} \to \makebox[\mylength]{$0$} \to \mathbb{Z} \to \mathbb{Z} \to
\mathbb{Z} \to 0
\end{gather*}
Now define a tower $\{ A_\alpha\}$ of ${\OC{G}{\frakF}}$-modules by
\begin{align*}
A_\alpha(G/1) & = 0
\\
\intertext{and for $H\in \mathfrak{F}\setminus\{1\}$}
A_\alpha(G/H) & = A_\alpha^{(n)}
\end{align*}
where $n\in \mathbb{N}$ is the unique index such that $H\leq H_n$. If $\varphi\: G/H
\to G/K$ is a morphism, then we set $A_\alpha(\varphi) = \id$ whenever
$A_\alpha(G/H) = A_\alpha(G/K)$ and otherwise we are forced
to set $A_\alpha(\varphi) =0$. Finally, the structure maps of
the tower $\{A_\alpha\}$ are the ones induced by the structure maps of the
towers $\{A_\alpha^{(n)}\}$.

Then the tower $\{A_\alpha\}$ satisfies the strict Mittag--Leffler condition in
the weak sense but it does not satisfy the strict Mittag--Leffler condition in
the strong sense.
\end{example}

The following statement is the key observation  on which Theorem A relies.
\fi

\begin{observation}
\label{obs:key-obs}
Let $\{A_\alpha\}$ be a tower of objects in $\mathfrak{A}^I$ which satisfies the
(strict) Mittag--Leffler condition in the weak sense. If $I$ has only finitely
many isomorphism classes of objects, then $\{A_\alpha\}$ satisfies the (strict)
Mittag--Leffler condition in the strong sense.
\end{observation}

\section{Proof of Theorem A}

The following, preliminary result  is essentially  \cite[p.~16, statement~3.1.3]{raynaud-71} translated to our setting. Since it needs some additional argument,   we include a detailed proof.

\begin{proposition}
\label{prop:raynaud-71}
Let $P$ be a contravariant ${\OC{G}{\frakF}}$-module. Let
\begin{equation*}
\ldots \leftarrow L_{4} \leftarrow L_{3} \leftarrow L_{2} \leftarrow
L_{1}\leftarrow L_{0}
\end{equation*}
be a co-tower of finitely generated free contravariant
${\OC{G}{\frakF}}$-modules such that $P=\varinjlim L_\alpha$.
Then the following statements are equivalent:

\begin{enumerate}
\item\label{prop:raynaud-71-1}
$P$ is projective.

\item\label{prop:raynaud-71-2}
The tower
\begin{equation*}
\ldots \to L_3^* \to L_2^* \to L_1^* \to L_0^*
\end{equation*}
satisfies the Mittag--Leffler condition in the strong sense.
\end{enumerate}
\end{proposition}

\begin{proof}
\ref{prop:raynaud-71-1} $\implies$ \ref{prop:raynaud-71-2}:  Since every projective is a direct summand of a free and by Proposition~\ref{lazard}, we can assume that $P$ is free on the ${\OC{G}{\frakF}}$-set $X$.
Now fix a $k\geq 0$ and assume that $L_k$ is finitely generated by the ${\OC{G}{\frakF}}$-set $Y_k$. Let $X_k \subseteq X$ be a $\OC{G}{\frakF}$-subset big enough so that the image of $L_k$ in $P$ is contained in $P_k$ the free $\OC{G}{\frakF}$-module generated by $X_k$. Note that $P_k$ is a direct summand of $P$ and that
 the canonical morphism $L_k \to P$ factors through  $P_k.$
 For each~$x \in X_k$ there is an~$i_x \geq k$ such that $x \in P$ is in the image of the canonical morphism $L_{i_x} \to P$. Now put  $i = \max\{i_x \, |\, x \in X_k\}$. Hence, for each $j\geq i$  the canonical map $L_k \to L_j$ factors through $P_k$.

Dualising now yields a surjection $P^* \twoheadrightarrow P^*_k$ and the map $L_j^* \to L_k^*$ factors through~$P_k^*$ such that $L_j^* \twoheadrightarrow P_k^*$.
Hence
$$\Im(L_j^* \to L_k^*) = \Im(P_k^* \to L_k^*)$$
is independent of $j$ and hence the claim follows.

\smallskip

\ref{prop:raynaud-71-2} $\implies$ \ref{prop:raynaud-71-1}:  Since $- \mathbin{\otimes_{\OC{G}{\frakF}}} M$ is a right exact
functor it follows  that the
tower $(L_\alpha^* \mathbin{\otimes_{\OC{G}{\frakF}}} M)$ of abelian groups satisfies
the Mittag--Leffler condition for any left ${\OC{G}{\frakF}}$-module~$M$. The natural
isomorphism $L_\alpha^*\mathbin{\otimes_{\OC{G}{\frakF}}} M \cong \Hom_{\OC{G}{\frakF}}(L_\alpha,
M)$ of Proposition~\ref{prop:dualising-bredon-modules} implies, that
$(\Hom_{\OC{G}{\frakF}}(L_\alpha, M))$ is a tower satisfying  the
Mittag--Leffler condition as well.

Consider the sequence of natural isomorphisms
\begin{equation*}
\Hom_{\OC{G}{\frakF}}(P, M)
\cong
\Hom_{\OC{G}{\frakF}}(\varinjlim L_\alpha, M)
\cong
\varprojlim \Hom_{\OC{G}{\frakF}}(L_\alpha, M)
\end{equation*}
which in turn yields an isomorphism
\begin{equation*}
\Ext_{\OC{G}{\frakF}}^1(P,M) \cong \varprojlim\nolimits^1 \Hom_{\OC{G}{\frakF}}(L_\alpha, M).
\end{equation*}
Since $(\Hom_{\OC{G}{\frakF}}(L_\alpha, M))$ is a tower of abelian groups which satisfies
the Mittag--Leffler condition, it follows that $\varprojlim\nolimits^1
\Hom_{\OC{G}{\frakF}}(L_\alpha, M)=0$, see for example \cite[Proposition~3.5.7]{weibel-94}. Therefore $\Ext_{\OC{G}{\frakF}}^1(P,M)=0$ and since this is
true for arbitrary~$M$ it follows that $P$ is projective.
\end{proof}

\iffalse
\begin{proposition}
\cite{gray-66}
\label{prop:gray-66}
Let $(A_\alpha)$ be a tower of countable abelian groups. Then, the following
conditions are equivalent:
\begin{enumerate}
\item The tower $(A_\alpha)$ satisfies the Mittag--Leffler condition.

\item $\varprojlim^1 A_\alpha = 0$.

\item $\varprojlim^1 A_\alpha$ is countable.\qed
\end{enumerate}
\end{proposition}
\fi

\iffalse
\begin{proposition}
\label{prop:cat-gray-66}
Let $(A_\alpha)$ be a tower of countable ${\OC{G}{\frakF}}$-modules. Then, the following
conditions are equivalent:
\begin{enumerate}
\item The tower $(A_\alpha)$ satisfies the Mittag--Leffler condition in the strong sense.

\item $\varprojlim^1 A_\alpha = 0$.

\item $\varprojlim^1 A_\alpha$ is a countable ${\OC{G}{\frakF}}$-module. \qed

\end{enumerate}
\end{proposition}
\fi

\begin{lemma}\label{lem:cat-gray-66} Let $(A_\alpha)$ be a tower of ${\OC{G}{\frakF}}$-modules, such that $A_\alpha(x)$ is countable for each $\alpha \in \mathbb{N}$ and $x \in {\OC{G}{\frakF}}$. Then the following
conditions are equivalent:
\begin{enumerate}
\item The tower $(A_\alpha)$ satisfies the Mittag--Leffler condition in the weak sense.

\item $\varprojlim^1 A_\alpha(x) = 0$ for every $x \in \OC{G}{\frakF}.$

\item $\varprojlim^1 A_\alpha(x)$ is a countable $R$-module for every $x \in \OC{G}{\frakF}.$

\end{enumerate}
\end{lemma}

\begin{proof} This was proved in  \cite{gray-66} for abelian groups.  The result now follows by applying that result to every $x \in \OC{G}{\frakF}.$
\end{proof}

\iffalse

\begin{proposition}
\label{prop:e-t-theorem-1.3}
Let ${\OC{G}{\frakF}}$ be a countable small preadditive category satisfying $(A)$ and
let $M$ be a countably generated flat right ${\OC{G}{\frakF}}$-module. Then, the following
conditions are equivalent:
\begin{enumerate}
\item $M$ is projective.

\item $\Ext_{\OC{G}{\frakF}}^1(M,R{\OC{G}{\frakF}})=0$.

\item $\Ext_{\OC{G}{\frakF}}^1(M,R{\OC{G}{\frakF}})$ is countable left ${\OC{G}{\frakF}}$-module.
\end{enumerate}

\end{proposition}

\begin{proof}
The proof of Theorem~1.3 in~\cite[p.~59]{emmanouil-12} should (with a few tiny
obvious technical modifications) prove also the above
Proposition~\ref{prop:e-t-theorem-1.3}.
\end{proof}

\begin{theorem}\label{thm:cat-et-thm-2.1} Let ${\OC{G}{\frakF}}$ be a countable small preadditive category satisfying $(A)$ and
let $M$ be a countable generated right ${\OC{G}{\frakF}}$-module of $\fld_{\OC{G}{\frakF}} M=n$. Then, the following
conditions are equivalent:
\begin{enumerate}
\item $\pd_{\OC{G}{\frakF}} M=\fld_{\OC{G}{\frakF}} M=n$.

\item $\Ext_{\OC{G}{\frakF}}^{n+1}(M,R{\OC{G}{\frakF}})=0$.

\item $\Ext_{\OC{G}{\frakF}}^{n+1}(M,R{\OC{G}{\frakF}})$ is countable left ${\OC{G}{\frakF}}$-module.
\end{enumerate}
\end{theorem}

\begin{proof}
If the statements in this section are all true, then the short proof of
Theorem~2.1 in~\cite[p.~60]{emmanouil-12} can be used
\underline{verbatim} to the generalisation.
\end{proof}

\fi

\begin{proposition}\label{propA}
Let ${\OC{G}{\frakF}}$ be of finite isomorphism type and
let $M$ be a countably generated flat contravariant ${\OC{G}{\frakF}}$-module. Then, the following
conditions are equivalent:
\begin{enumerate}
\item\label{propA-1} $M$ is projective.

\item\label{propA-2} $\Ext_{\OC{G}{\frakF}}^1\Bigl(M; \prod\limits_{x\in {\OC{G}{\frakF}}_0} \mathbb{Z}[-,x]\Bigr)=0$.

\item\label{propA-3} $\Ext_{\OC{G}{\frakF}}^1\Bigl(M; \prod\limits_{x\in {\OC{G}{\frakF}}_0} \mathbb{Z}[-,x]\Bigr)$ is a countable abelian group.
\end{enumerate}
\end{proposition}

\begin{proof}
As before we have that \ref{propA-1} $\implies$ \ref{propA-2} $\implies$ \ref{propA-3}. To show \ref{propA-3} $\implies$ \ref{propA-1}
we follow the same outline as the proof of \cite[Theorem 1.3]{emmanouil-12} with some adaptation to the setting of ${\OC{G}{\frakF}}$-modules.
Since $M$ is countably generated and flat, it is the direct limit of finitely generated free modules $L_\alpha,$ for $\alpha \in D$ some countable directed set.
By \cite[Application 3.5.10]{weibel-94}, see also the remark before that Application,  and the fact that the $L_\alpha$ are projective, we have
$$\Ext_{\OC{G}{\frakF}}^1(M; \prod\limits_{x\in {\OC{G}{\frakF}}_0}\mathbb{Z}[-,x]) \cong \varprojlim\nolimits^1 \Hom_{\OC{G}{\frakF}}(L_\alpha, \prod\limits_{x\in {\OC{G}{\frakF}}_0}\mathbb{Z}[-,x]).$$
In particular, $ \varprojlim\nolimits^1 \Hom_{\OC{G}{\frakF}}(L_\alpha, \prod\limits_{x\in {\OC{G}{\frakF}}_0}\mathbb{Z}[-,x])$ is a countable abelian group and hence
Lemma \ref{lem:cat-gray-66} implies  that the inverse system $\Hom_{\OC{G}{\frakF}}(L_\alpha, R{\OC{G}{\frakF}})=L_\alpha^*$ satisfies the Mittag--Leffler condition in the weak sense. But since ${\OC{G}{\frakF}}$ is of finite isomorphism type, it satisfies the Mittag--Leffler condition in the strong sense.

Finally apply Proposition \ref{prop:raynaud-71}.
\end{proof}

\smallskip\noindent{\bf Proof of Theorem A.} This now follows directly from Proposition \ref{propA} by an easy dimension shift. \qed

\section{Groups of Bredon-homological dimension $1$}\label{Bredonhom-1}

In this section we will consider Bredon-cohomology for the family of finite subgroups only, and  we will  formulate a Bredon-analogue to Bieri's conjecture that every group of homological dimension $1$ is locally free, or equivalently, that every finitely generated group of homological dimension $1$ is free. The naive analogue to this conjecture is to ask whether every finitely generated group of Bredon-homological dimension $1$ is virtually free, but this is not the case as the lamplighter group shows~\cite{bridson-12}. Hence a rational version of Bieri's conjecture would need some further restrictions as well.
We propose a somewhat stronger assumption than finite generation, and suggest that the Bredon-analogue to finite generation is being of type Bredon-$\FP_1$.

To stay in line with convention, we shall, in this section, denote the Bredon finiteness conditions by underlining. In particular, we put $\hd_{\frakF_{\fin}} G = \uhd G,$ $\cd_{\frakF_{\fin}} G = \ucd G,$ and denote by $\ul{\FP}_n$ the condition Bredon-$\FP_n$ for the family $\mathfrak{F}=\frakF_{\fin}.$

We ask the following question and devote the rest of the paragraph to giving some evidence for a positive answer, see Theorem \ref{hdf1}.

\begin{question}\label{conj:bredon-bieri}
Let $G$ be a group of type $\ul{\FP}_1$ and with $\uhd G=1$. Is~$G$  virtually free?

\end{question}

Note that for torsion-free groups this question is exactly the question posed by the finite generation version of Bieri's original conjecture. It immediately follows from \cite{dunwoody79} that an infinite virtually free group has $\hd_\mathfrak{F} G =1$. Using Bass--Serre theory and a result by Karras, Pietrowski and Solitar, Dunwoody actually shows that a finitely generated group has $\cd_\mathbb{Q} G=1$ if and only if $G$ contains a free subgroup of finite index \cite[Corollary 1.2]{dunwoody79}. Furthermore, see \cite[Theorem 1.1]{dunwoody79}, a group~$G$ has $\cd_\mathbb{Q} G=1$ if and only if $G$ acts on a tree without inversions and with finite vertex stabilisers. This, in turn, implies that $\cd_\mathbb{Q} G =1$ if and only if $\ucd G =1.$

Also note that Question \ref{conj:bredon-bieri} has a positive answer for elementary amenable groups: they have Hirsch-length $1$, hence are locally-finite-by-(virtually torsion-free soluble) \cite[(g)]{wehrfritz}. Now the $\ul{\FP}_0$ condition implies that there is a bound on the orders of the finite subgroups. Hence the group is finitely generated and finite-by-virtually torsion-free soluble of Hirsch-length $1$ and thus  is virtually infinite cyclic.

\begin{remark}\label{lnremark}
In the above question one can  weaken the hypothesis of $\ul{\FP}_1$ to only asking for the group to be finitely generated and of type $\ul{\FP}_0$. There are examples of finitely generated groups (even $VF$-groups) that are of type $\ul{\FP}_0$ but not of type~$\ul{\FP}_1$, see \cite[Example 4]{ln-03}. These examples are, however, of Bredon-homological dimension at least $2$. Hence the question arises whether there are such examples of Bredon-homological dimension $1$.
\end{remark}

\begin{lemma}\label{coherent-lem}
Let $G$ be a virtually free group. Then the group ring $\mathbb{Z} G$ is coherent.
\end{lemma}

\begin{proof}
Since $G$ is virtually free, it has a finite index free subgroup $N$, whose group ring $\mathbb{Z} N$ is coherent. Now suppose we have a finitely generated $G$-module~$M$. Restriction to a finite index subgroup preserves finite generation, hence, as an $N$-module, $M$ is finitely presented. In particular, consider the short exact sequence of $G$-modules
$$ 0 \to K \to F \to M \to 0,$$
where $F$ is finitely generated free. As an $N$-module $K$ is finitely generated. Hence, the induced module $K \otimes_{\mathbb{Z} N} \mathbb{Z} G \cong K \otimes \mathbb{Z}[G/N]$ with the diagonal $G$-action is a finitely generated $G$-module. We also have a $G$-module epimorphism $\mathbb{Z} [G/N] \twoheadrightarrow \mathbb{Z}$. Therefore $K \otimes \mathbb{Z}[G/N] \twoheadrightarrow K,$ and $K$ is finitely generated as a $G$-module as required.
\end{proof}

Following \cite[Section 5]{emmanouil-12} we denote by $\frak S$ the class of groups for which the group ring $\mathbb{Z} G$ is a left $S$-ring; that is to say that every finitely generated flat $\mathbb{Z} G$-module is projective. We denote by $v\frak S$ the class of all groups containing a finite index subgroup belonging to $\frak S$.

Similarly, we define the class $\underline{\frak S}$ to be the class of groups for which every finitely generated flat ${\OC{G}{\frakF}}$-module is projective.

\begin{theorem}\label{hdf1}
Let $G$ be a group of type $\ul{\FP}_1$ and of $\uhd G=1$. Then the following are equivalent.

\begin{enumerate}
\item\label{hdf1-1} $G$ is virtually  free;
\item\label{hdf1-2} $G$ is virtually residually torsion-free nilpotent;
\item\label{hdf1-3} $G \in v\frak S;$
\item\label{hdf1-4} $G\in\underline{\frak S}$.
\end{enumerate}
\end{theorem}

\begin{proof}
$\ref{hdf1-1}\iff \ref{hdf1-2} \iff \ref{hdf1-3}$ follows directly from \cite[Proposition 5.5]{emmanouil-12}.

$\ref{hdf1-1} \implies \ref{hdf1-4}$: By \cite[Lemma 3.1]{kropholler-09} it follows that for a group  of type Bredon-$\FP_n$,  the Weyl-group $WK=N_G(K)/K$ is of type $\FP_n$ for all $K \in \mathfrak{F}$. Since $G$ is virtually free, it follows that $WK$ is virtually free as well. Hence by Lemma \ref{coherent-lem} the group rings $\mathbb{Z}(WK)$ are coherent. Now let $M$ be a finitely generated flat ${\OC{G}{\frakF}}$-module. Hence $M(G/K)$ is a finitely generated $\mathbb{Z}(WK)$-module. Since $\mathbb{Z}(WK)$ is coherent, it follows that $M(G/K)$ is finitely presented for all $K \in \mathfrak{F}$. Since $G$ is, in particular, of type $\ul{\FP}_0$, \cite[Lemma 3.2]{kropholler-09} implies that $M$ is a finitely presented flat ${\OC{G}{\frakF}}$-module, and hence is projective \cite[Corollary 3.3]{nucinkis-04}.

$\ref{hdf1-4} \implies \ref{hdf1-1}$: Since $G$ is of type $\ul{\FP}_1$ we have  a short exact sequence:
$$0\to M(-) \to \prod_{K \in \mathcal{F}}P_K(-) \to \mathbb{Z}(-) \to 0.$$
with  $\mathcal{F}$  finite and $M(-)$  finitely generated. Since $\uhd G=1$, $M(-)$ is also flat. Now~$\ref{hdf1-4}$ implies that $M(-)$ is projective and hence $\ucd G =1$. This implies that $\cd_\mathbb{Q} G = 1.$
Using the fact that $G$ is finitely generated we can now apply \cite[Corollary 1.2]{dunwoody79} to get the claim.
\end{proof}

\section{Groups of rational homological dimension $1$}

In this final section we shall present a straightforward  generalisation of \cite[Proposition 3.2]{emmanouil-12}.

\begin{lemma}\label{countable-q-lem}
Let $G$ be a finitely generated group and $M$ be a countable $\mathbb{Q} G$-module. Then
$H^1_\mathbb{Q}(G, M)$ is a countable abelian group.
\end{lemma}

\begin{proof} The proof is completely analogous to that of \cite[Lemma 3.3]{emmanouil-12}.
\end{proof}

\begin{theorem}
Let $G$ be a finitely generated group of $\hd_\mathbb{Q} G=1$. Then the following are equivalent:
\begin{enumerate}
\item\label{thrm-p1} $G$ is virtually free.
\item\label{thrm-p2}$G$ has a non-trivial finitely generated free group as a normal subgroup.
\end{enumerate}
\end{theorem}

\begin{proof} \ref{thrm-p1} $ \implies$ \ref{thrm-p2}: It is a well known fact that if a group is virtually of property~$\mathcal P$, where $\mathcal P$ is a property that is closed under taking finite index subgroups, then there is a normal subgroup $N$ of finite index possessing this property $\mathcal P$.

\smallskip\noindent \ref{thrm-p2} $\implies$ \ref{thrm-p1}: It suffices to show that $H^2_\mathbb{Q}(G, \mathbb{Q} G)$ is a countable abelian group. An easy dimension shift applying Theorem \ref{main-et} together with Duwoody's result \cite{dunwoody79} then yields the claim.

Consider the Lyndon--Hochschild--Serre spectral sequence:
$$H_\mathbb{Q}^p(G/N, H_\mathbb{Q}^q(N, \mathbb{Q} G)) \rightarrow H_\mathbb{Q}^{p+q}(G, \mathbb{Q} G).$$

Since $N$ is free, we have that $H^q(N, \mathbb{Q} G)=0$ for all $q\geq 2$. We also claim that $H^0_\mathbb{Q}(N, \mathbb{Q} G)=0$: Since $N$ is finitely generated free, it is of type $\FP_\infty$ over~$\mathbb{Q}$, and hence it suffices to show that $H^0_\mathbb{Q}(N, \mathbb{Q} N)=0$, which follows from \cite[Proposition 13.2.11.]{ross08}.

 This now implies that
$$H_\mathbb{Q}^2(G, \mathbb{Q} G)= H^1(G/N,H^1(N, \mathbb{Q} G)).$$
The result now follows from Lemma \ref{countable-q-lem}.
\end{proof}

\iffalse
\vfill\eject

redpajama_set_name: RedPajamaArXiv


Ahead of the 11th parliamentary election slated for December 30, candidates are campaigning to persuade voters to vote for them.
There are only six days left for the election, but in Dhaka 10 constituency, only the Awami League candidate,Barrister Sheikh Fazle Noor Taposh, can be seen campaigning on the field in full swing.
In Dhaka 10, there are six candidates contesting the election: Abdul Mannan from BNP with the symbol of "sheaf of paddy"; Abdul Awal from Islami Andolon Bangladesh with "handfan"; Bahrane Sultan Bahar from Progotishil Gonotantrik Dal with "tiger"; Helal Uddin from Jatiya Party with "plough" symbol; and Sheikh Fazle Noor Taposh from Awami Leauge with "boat" symbol.
Abdul Awal can also be seen campaigning sometimes in this constituency. The rest of the candidates cannot be seen anywhere in public.
Bangabandhu Sheikh Mujibur Rahman's grandnephew, Sheikh Taposh, has been a lawmaker from Dhaka 10 for two consecutive terms.
Dhaka 10 consists of four police stations _ Dhanmondi, Kalabagan, New Market and Hazaribagh with six wards.
While visiting the areas,this correspondent has seen posters of the Awami League candidate abound, and very few of Islami Andolon Bangladesh. It is rare to find posters of BNP and other candidates in the area.
Lawmaker Sheikh Taposh told the Dhaka Tribune that every party has its own strategy regarding campaigning.
"We are campaigning in our own way, but we do not know why BNP and the other parties are not campaigning," he said.
"We do not even restrict any form of campaigning by imposing any barriers or obstructions from our side," he added.
Meanwhile, BNP Chairperson's Press Wing Member Shamsuddin Didar said: "How can we campaign when there is no level playing field in the election?"
"Everywhere, police with the support of the ruling party,are keeping us under pressure and are continuously oppressing us," he said.
BNP sources said its candidate for Dhaka 10,Abdul Mannan, will start his campaign from tomorrow,with a large showdown from the Dhaka City College premises around 3:30pm.
"About three to four thousand people will participate in the showdown," sources added.
Tarek Shikder, Awami League candidate Sheikh Taposh's media and public relation coordinator, said Awami League has been campaigning since December 10.
"We have already campaigned three times in every ward, and will continue to do so till December 27," he added.
Md Salam, a shopkeeper of Green Road area, said:"We want to see Taposh Bhai for another term and will vote for him. He is very close to us."