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Annual Report for the year ending 31 December 2016
January 26, 2017 /in News /by m7alpha
If you're interested what the achievements and challenges of Dr Klown were in 2016, read the Report below.
Download: Annual Report 2016 publication
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Teambuilding Event
December 5, 2016 /in News /by m7alpha
All Dr Klown volunteers gathered to discuss possibilities to improve the services delivered to children in hospital and ways to enhance co-operation and the smooth running of the Association. Many ideas were brought forward that will be implemented over the coming year.
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Chris Mamo's Marathons Project 20-20-20
August 19, 2016 /in News /by m7alpha
We can finally, and proudly, an nounce that the Project 20-20-20 has finally reached completion and that 20 beds have been installed in the paediatric wards that will benefit a number of parents of hospitalised children by giving them better sleeping
conditions whilst caring for their children.
During the inauguration of the beds at Mater Dei Hospital, Dr Klown President, Maurice Sleypen, said: "We went to see the beds installed in the rooms and found most of them in use already. Not just during the night, because most parents sleep
according to the rhythm of their child. The many mothers we spoke to were very happy with the comfort of the beds as opposed to the chairs which caused neck pain and backaches."
On the other hand, Chris Mamo, the person behind the project, stated that "It's so gratifying to hear how very welcome those beds are with the parents who have to stay long nights in their child's room. It has definitely been worth all the sacrifices and pain. I would do it all over again!"
The Project was a collaboration between Chris Mamo, a Maltese/English personal trainer who previously served 12 years with Her Majesty's British Forces, and Dr.Klown, an NGO active in Mater Dei Hospital (Malta), that provides hospitalised children distraction and stress relief through fun and laughter. Dr Klown and Chris Mamo joined forces in a project to provide the parents of hospitalised children with a comfortable foldable night-bed in the sick child's room instead of sleeping upright in a seat all night and waking up with neck pain and backaches. Chris Mamo ran 20 consecutive marathons of over 42km from the 19th of March, with the support of Dr. Klown and their troupe of volunteers.
18 beds were installed at Mater Dei Hospital, while another 2 beds have been installed at the Gozo General Hospital. Dr Klown will be extending its services to Gozo as from September 2016. A call for new clown doctors resident in Gozo has been made, and anyone interested in joining Dr Klown should express their interest by applying at www.drklown.org/contact/
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Specialised Clown Doctor training
April 20, 2016 /in News /by m7alpha
The troupe of Klown Doctors has increased to reach 26 in November 2015 after an initial training programme in artistic skills and basic hospital clowning competences for the new recruits. The programme was led by Ms Laurence Quetel, a seasoned clown trainer and herself a hospital clown working with the Cliniclowns in The Netherlands and Belgium.
The next step in the continuous improvement of our services is a live in seminar on 23-24 April 2016, when we will focus on Clown Logic, Working in diads / triads and develop short sketches to become ever more proficient as Klown Doctors.
There will be concurrent sessions with 2 trainers, Ms Laurence Quetel and Mr Sean Buhagiar, because our troupe has grown too big (26 Klowns) to train properly as one group.
Both programmes of November 2015 and April 2016 are to a great extent made possible by funds of the Malta Community Chest Fund.
Malta Community Chest Fund
First Call – Social Projects
Projects part-financed by Malta Community Chest Fund
Co-financing rate: 90% MCCF funds. 10% Beneficiary's funds
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Successful completion of Development programme
A programme to enhance the hospital clown skills of the Klown Doctors who visit hospitalised children in Mater Dei every week, was successfully concluded.
The programme started in March 2015 with a live in training session spanning over 2 days (Saturday 7 and Sunday 8 March) and was followed by Peer session meetings to integrate the newly acquired competences in their regular visits. The programme was funded by the Small Initiatives Support from the Malta Council for the Voluntary Sector.
Klown Doctors discovered during their latest seminar that there's quite a difference between Klown Logic and Common Sense!
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Project 20-20-20 a great success!
20 Marathons in 20 consecutive days for 20 beds was the challenge that Chris Mamo took upon himself and successfully brought to an end under enthusiastic applause by the Klown Doctors on April 7.
This was a joint project between the Dr Klown team and Chris Mamo, who was supported by Jonathan Azzopardi (Springbox Media) for the marketing of this campaign and friends and family for support and logistics. Where needed – and towards the end of the challenge this meant every day! – the staff of MDH Physiotherapy Department helped Chris get ready to endure another day of pain.
Chris was encouraged every day by volunteers of Dr Klown at start and finish and was regularly accompanied by other runners for part of the route, especially by the Lady Runners Club.
The goal of providing 20 beds for the parents who need to stay overnight in the room of their child in hospital has been achieved! Even better: Dr Klown and Chris are now figuring out how best to invest the extra donations received in line with the original goal of beds for parents of hospitalised children.
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More Common Sense
The sequel to the successful book Common Sense, written by dott. Edward Curmi, was launched by H.E. the President Marie-Louise Coleiro Preca at the Palace in Valletta on April 6.
During the ceremony, Dr Anthony Grech compared common sense with scientific research and highlighted the differences and similarities between both.
Ms Francesca Balzan pointed out the value added by illustrations to the message of a book and particularly the intelligence and beauty of the works by artist Marisa Attard that illustrate More Common Sense, which were exhibited in the Palace Dining Room for the evening.
Her Excellency drew attention to the need for common sense in society, especially in these confusing times. She also praised the NGO Dr Klown for it's relentless endeavour to relieve pain and suffering of hospitalised children and their parents.
The proceeds of Dott. Curmi's book as well as of the sale of the framed illustrations by Marisa Attard are in support of Dr Klown NGO.
The book is available at all Agenda outlets.
The illustrations / paintings of Marisa Attard are for sale through the artist. Contact Marisa for viewing on [email protected].
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VOCA in Concert
March 21, 2016 /in News /by m7alpha
Voca Choir is a group of fun-loving, professionally-minded singers and musicians, who share a love for song. Since its inception, Voca Choir have made it their primary mission to not be just any "choir", but a choir with a difference. Today, Voca Choir's repertoire ranges from spiritual and worship music, to folk, musicals and also contemporary pop, although Voca Choir's distinguishing forte is Gospel music.
This makes them one of the top Maltese Gospel Choirs performing both locally and overseas. This year, Voca Choir have joined forces with MSV Life to organise a funpacked musical concert on Friday 29th April 2016, at Santa Monica School Theatre, Fleur-de-lys, B'Kara in aid of Dr Klown. Voca Choir will be performing a mix of musical styles from its vast repertoire, namely pop songs and gospel music as well as some classical pieces – including covers from Prince of Egypt, Celine Dion, Cold play and many more. Join us for this musical evening by booking tickets on www.ooii.com.mt
MSV Life Voca Choir in Concert – Press Pack 2016
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The second book of Dott. Edward Curmi, entitled MORE COMMON SENSE, will be launched at San Anton Palace the 6th of April by H.E. the President. The book's illustrations are by Ms Marisa Attard and these will be exhibited during the book launch. All proceeds of the book and a major part of the sales of the framed original illustrations will go towards the NGO Dr Klown, thanks to the generosity of both the author and the artist.
The book, containing lots of practical psychological advice, will be for sale at all Agenda Bookshops and directly from Dr Klown at [email protected].
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20 Marathons in 20 Days for 20 Hospital Beds
It was last summer whilst out running with a family member, that I knew I wanted to make a positive change somehow. It felt right to give back by doing something I had always done even as a child, run. Since I made the decision that day, many things have changed. Do I run a distance, do I come up with a new challenge, or would I take on an existing event? Eventually, I decided to run a number of full marathons, 42.195km. How many had yet to be decided. What I did know was, that it would be for a local cause, and that it would benefit children in some way.
I'm a 33-year-old Anglo-Maltese personal trainer who previously served 12 years with Her Majesty's British Forces. I have always kept fit and healthy to some degree and pride myself in helping others realize their potential health and fitness goals. I have been living in Malta for four years now. I am engaged to the love of my life Natassja, and a stepfather to my precious princess Megan.
The search was on for a charity I felt would benefit from my event. It didn't take long before I found DrKlown. I new very little about the NGO until I met Maurice and Christine, the president and vice president of DrKlown here in Malta. The more I learnt about the fantastic work they and the clowns did in the pediatric wards of Mater Dei Hospital, the clearer it became that this was an organization I wanted to do something for, as through them I could make a difference.
We immediately understood that our goal was similar, we both wanted to help children and bring happiness even in the most difficult of times. After visiting the pediatric wards, which was my first time, I was astonished at the staff's level of professionalism, dedication, and courage that went into the daily running of the wards. After talking with the staff, we asked if they needed anything we could help with, or whether there was something that could be improved in the wards to make their lives easier. I found that the response we got was particularly selfless and kind, that is, to ask not for themselves but for the parents of the hospitalized children. They wanted to help the parents by giving them better sleeping conditions when they had to sleep at the hospital.
Having one of your children hospitalized for a length of time is hard enough for any parent, but adding restless nights does make the situation even more difficult than it already is! It was clear that parents spending the night, a week, or a few months with their hospitalized child would benefit from wall-mounted pull-down beds.
This has to be a place where they could get proper rest, giving them time to recharge and have the strength to face whatever the next day might bring. This in turn would give the children what they need from their parent; hope and courage to deal with whatever situation they were in.
We calculated that between the three pediatric wards we would need a total of 20 beds. In that way, all single rooms would get their own bed.
From that point on my task became clear. I would be running 20 Marathons for 20 beds.
Everyone said it was crazy to run 20 consecutive marathons. I think it's crazy not to help someone when you truly believe you can.
Moreover, I have a strong belief that we are all capable of giving back to the community in some way. This could be a small gesture of kindness to a neighbour or friend, a decision to impact the people around us in a positive way, or making a choice to be the best version of ourselves we can be.
When this challenge succeeds, I hope it will encourage people to realize that no matter where we come from, what chain of events led to this point in our lives, or even how we think we are perceived by others, not only can we change our lives for the better but we also have the ability to change the lives of others!
In an age of uncertainty, and perhaps sometimes negativity at every turn, I'm just a guy that wants to make a positive change in my own way. With your help, I know this change is possible. With your help, we know we can achieve this goal and make a difference. You can do your part by donating, and helping us spread the message.
Donations Links
Donate €5 – http://bit.ly/1Qr2BJZ
Donate €10 – http://bit.ly/1WxD9ak
Donate €20 – http://bit.ly/1TvVYg8
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Revised edition. Hardcover, leatherette, 378 pp. w/ 51 b&w illustrations and manuscript photos. Biography of Shakespeare based on the documents in the historical record. Book Condition : Very Good; no text marks; stain on side text block; slight crease to front cover. No Dust Jacket as issued.
Limited Edition. Romantic history with short biographical sketches of 30 colonial men and (primarily) women including Abigail Adams, Martha Washington, Eliza Custis, and more. Includes 61 b&w illustrations. Book Condition ; Good-; no text marks, former owner's bookplate inside front cover; binding relaxed; age toning to page edges; bumping to spine ends; darkening to spine. No Dust Jacket as issued.
83 Dante-Handbuch. Einführung in das Studium des Lebens und der Schriften Dante Alighieri's Scartazzini, G.A.
4th Edition 1977 (reprint of 3rd edition). Biography of the Silesian reformer whose following includes several Schwenkfelder congregations near Philadelphia. Book Condition : Very Good; no internal markings; light rubbing to covers; small crease at top right front cover corner; larger horizontal crease at front cover/spine; price sticker remnant at bottom right rear.
A biography in 2 volumes of the American Romantic poet, critic, editor, and diplomat James Russell Lowell. 1st edition. Black cloth boards, gilt spine titles, top edge gilt, vol. 1 - 455 pp., vol. 2 - 482 pp.; 12 black and white illustrations (6 in each volume, including a portrait frontis.). PLEASE NOTE : Extra postage may be required due to weight of volumes. Condition: Good, ex-seminary library, nicely taken care of, rear pocket removed, spine labels, stamps inside cover, penciling on t.p. verso, pencil marks in margins very rare.
86 Schwenkfeld Knight of Faith Seyppel, Joachim H.
Subtitled " A Study in the History of Religion". Examines the theology of the Silesian reformer whose following includes several Schwenkfelder congregations near Philadelphia. Attempts to explain his ideas and compare them to his peers and later theologians. Book Condition : Good, Ex-Seminary Library; no text marks; bookplate inside front cover; catalog labels on bottom spine ad top right rear; pocket removed from inside rear cover tearing paper; light rubbing to covers, light bumping to spine ends and cover corners; a few spots to text block.
Inscribed by Author on back of title page : "For Frances and Richard with appreciation and affection, Cornelia 10/15/'81". A monograph biography of Robert Browning concentrating on his family background, early writings, and romance and marriage to Elizabeth Barrett. Limited First Edition No. 150/1000. Book Condition : Near Fine; minimal wear. Dust Jacket Condition : Very Good; some shelf wear to top edge.
89 Goethe's Life-Poem - As set forth in his life and works Snider, Denton J.
A biography of Johann Wolfgang von Goethe, German author and poet. 1749-1832. Steel blue cloth boards, gilt spine titles, 601 pp., unillustrated. Condition: Good-; mild wear to corner tips, spine ends, a nick on front fore edge. a 2-inch split of paper along inner hinge, front cover loose but secure.
This pamphlet, part of the British Council's series "Writers and Their Work", is a brief biography of noted Victorian essayist Walter Bagehot. Editior-In-Chief of The Economist, Bagehot wrote extensively on economic, government, and society as well as literary and poetic criticism. Import. Book Condition : Very Good; slight age discoloration; small crease at right bottom front cover; no internal markings.
Edited collection of Civil War letters. Stanard was one of the Virgina Military Institute cadets pressed into service as well as one of the fatalities of the famous VMI charge of the Federal ranks during the Battle of New Market in May 1864. Book Condition : Good; no text marks; spotting to text block; foxing to free end papers. Dust Jacket Condition : Good; wear/chipping to top edge; rubbing.
92 Safir (Ambassador) Steeves, John M.
Inscribed by Author on front free end paper : "To John & David Dorman, Here's to Old times and Wonderful Memories, Cordially John Steeves, Hershey '95." Autobiography by the former Director General of the Foreign Service details his life especially his diplomatic service including serving as Ambassador to Afghanistan (1961-66). Book Condition : Fine; printing label inside front cover. Dust Jacket Condition : Very Good; minor wear to edges; some light rubbing.
Commemorating the Eisenhower Centennial (1890-1990), this edited booklet contains excerpts from "At Ease: Stories I Tell to Friends", Eisenhower's presidential papers; and the Official Records from the War of the Rebellion (Civil War). Book Condition : Good-; no text marks; some spotting to covers; price sticker remnant/slight tear at bottom left front; crease at bottom right front cover effecting first 1/3 of book.
94 Keystone of Justice: The Pennsylvania Superior Court Tamilia, Patrick R.; Hare, John J.
Diary of early 19th century Quaker merchant and shipowner in Philadelphia provides a unique social history of the time and place, as well as the author's views on democracy, wealth, women's rights, and Quakerism. 22 b&w illustrations. Book Condition :Good; no text marks, owner's name on half title page; page tip creasing; light bumping to spine ends and cover corners; some staining to text block. Dust Jacket Condition : Fair; chipping and tearing to edges, corners, and spine ends; overall rubbing; price unclipped.
Signed by Author on title page. Biographical sketches of 6 Generals and 2 Admirals from Columbia Pennsylvania - Thomas Welsh, Edward Shannon, Daniel Strickler, Charles Supplee, Richard Snyder, Frank Smoker, Charles Mason, and Dr. Richard Kern. Privately Published. Book Condition : Very Good; light rubbing to covers; curling to front cover edge; a few spots to text block.
1927 New and reset edition w/ additional material. 477 pp. includes several b&w plates. Autobiographical writings including Reveries Over Childhood and Youth (1914); The Trembling of the Veil (1922-1926); and notes. Book Condition : Good; no text marks; light bumping to spine ends; toning to end papers from inlaid DJ; owner's name on front free end paper. Dust Jacket separated into 4 pieces and inlaid.
100 Festschrift Friedrich Carl Andreas zur Vollendung des siebzigsten Lebensjahres am 14. April 1916: Dargebracht von Freunden und Schülern Wackernagel, J. ; H. Oldenberg ; A. Debrunner ; et al.
Softcover, protected by seminary library pressboard wraps, with original spine visible. 142 pp, 2 plates (1 color). Text in German. Probably first edition. 14 articles, each with a different author, on Andreas and his work. Condition: Good; ex-seminary library with typical markings, labels, stamps and residual from removed rear pocket on pressboard cover, stamps and penciled notes on extremities. Reading pages are clean, edges crisp, some creasing. Age darkening around edges of covers, original spine largely missing, but title visible. One piece separated, laid in.


Gregor everytime you post one of the "cheesy" girl shots I think less of you.
Are you twelve years old and have never seen a woman before? Before too many of you stand up in arms against this post, think about it… really, why this fluff on a hockey website?
If you don't like the girls feel free to scroll past her or you could always ask for a refund.
OldTimer. You're giving old people a bad name!
If fluff bothers you take a Reactin.
The question should be….why do we keep bringing up hockey on this Fluff website?
You can debate the "why", but the "what" is apparent … the Oilers are clearly playing their best hockey of the season and their final 3rd of the season looks to be their best by a significant margin!
I can tell you it isn't.
Last year the Oilers crushed Calgary on April 3rd and were tied for 8th.
They then lost 9 of 10 and dropped out of the playoffs, before winning their final two games.
They won 3 of their final 12 (quarter of the season) so they did not end the season strong.
In 2012, they went 7-9-4 in final 20 games.
In 2011, they went 5-11-4 in final 20 games.
In 2010, they went 8-10-2 in final 20 games.
Honestly Jason, I think Klefbom playing the right side has less to do with Fraser and is more about the draft. I think the organization sees Marincine, Klef, and Nurse all on the left side and is looking at Ekblad for the right side. I think if Klefbom fits nicely on the right side, it helps bring more balance to the RD/LD down the road and allows us a little more freedom to choose a center this summer. That the way I read it at least.
It will end up being the best as well. Incredible what the team can accomplish with poor d, no two way play from the forwards, and average to great goal tending.
However, this team is going to get eaten alive when their schedule turns back to the pacific division against teams all jockeying for pole position.
I hope we get some wins against teams the Oilers should be able to beat, against teams that at the beginning of the season, everyone said we should have no problem with. However, the true measuring stick of how much work this team will need to do in the off season will be against the big pacific teams coming down the pipe.
They stink, and its a ridiculous sample size. Eventually it evens out.


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Merry Wives of Windsor tour 2018
23 May - Perranporth: Memorial Hall (changes to schedule made 27/11/17)
24 May - St Austell: Arts Centre (changes to schedule made 27/11/17)
30 May - Truro: Burrell Theatre
31 May - Penzance: Zennor Theatre, Penwith College
2 June - Bude: Parkhouse Centre
7 June - Falmouth: Princess Pavilion
Booking has not yet opened - watch this space.
Opera World Premiere in Cornwall next week!
On Friday 10th November the world premiere of a new opera/oratorio by Stephen McNeff based on a libretto by Charles Causley will be performed at Launceston Town Hall and on the following day 11th November in the Guildhall, St Ives. The new opera/ oratorio has been commissioned by the Bournemouth Symphony Orchestra based on Causley's The Burning Boy, a libretto written some years before his death that remains one of the most important, but as yet unperformed, Causley texts. BSO ensemble Kokoro, which specialises in promoting the works of twentieth and twenty-first century composers, will perform the new opera in two Carn to Cove promotions in Cornwall. This follows BSO and Carn to Cove's highly successful residency day which takes place every year when the orchestra performs three sectional concerts across Cornwall in Carn to Cove's village network.
The Burning Boy tells the story of the prophet Elisha and the raising of the boy from the dead after he has been struck down in the field by the heat of the sun. The source is biblical but Causley transposes it to the West Country at an unspecified time and blends it with the ritual of the harvest and the myths of the corn spirits.
The performance will bring together both amateur and professional musicians, as well as local Cornish school children. BSO Associate musicians in Cornwall, Patrick Bailey and Matt Harrison, are working with local schoolchildren to form a children's chorus for the project. Paul Carey Jones the Welsh baritone who made his Royal Opera House debut last year takes the role of Elisha. For many of the children involved, this will be a once-in-a-lifetime experience. Talented music students from Trinity Laban Conservatoire of Music and Dance are also included in the company working with the BSO training programme.
Stephen McNeff has worked exclusively as a composer for over 40 years and is one of the UK's most successful composers of works bringing professional and amateur talent together..
Talking about the BSO commission, Stephen McNeff said, "I am delighted to be working on The Burning Boy in Charles Causley's centenary year, Charles and I had a long and fruitful working relationship over many years and The Burning Boy was a libretto he wrote especially for me. I am also thrilled to be working once again with the BSO and Kokoro, and having my new work premiered fittingly in this centenary year on the Carn to Cove Cornwall rural touring network".
10th Nov @ 7:30pm
Launceston Town Hall| tickets from £5 - call 01566 772321|www.carntocove.co.uk/events
St Ives Guildhall |tickets from £5 - call 01726 897500|www.carntocove.co.uk/events

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: How to quickly jump to the beginning of a command on terminal? For example:
% grep my_words /xx/xx/xx/xx/xx/xx/xx/xx/name.txt

When I am the end of the command line, I often need to jump to the beginning, i.e. before grep or after the grep command. I have to move backward character by character, and it's very slow.
How to do that?

A: You can simply click the home key to move the cursor to the start of the command.

redpajama_set_name: RedPajamaStackExchange


My husband, Tim, pointed out that I sounded like that guy from Airplane. Remember? "I picked a bad week to give up drinking...I picked a bad week to give up amphetamines...I picked a bad week to stop sniffing glue..."
Well, that made me laugh. The comments from friends on Facebook made me laugh too. And all the laughing made me forget how cold I was (well sort of).
I wore this cozy hat all day, but lost the dopey grin right after this weird picture was taken.
And then, instead of sulking and feeling all mad at the world at the unfairness of it all, we danced. We danced to stay warm. We danced to have fun. We danced to say screw you to the world and the unfairness of it all (okay maybe that was just me).
Here it is, our Kitchen Dance #547- The Broken Furnace Edition.
Furnace update: It has been fixed (we hope) and we are all safe, warm and happy.
Oh my gosh, I love your family so much. And I'm so glad you got your furnace fixed (hopefully).
I'm serious and don't call me Shirley!


text:
Despite Modest Gains, Women and Minorities See Little Change in Representation on Fortune 500 Boards [REPORT]
TOPICS:BusinessMulticultural
Stock Market and Exchange, Data, Graph, Chart, Shanghai
Posted By: Gene February 5, 2018
Boardroom diversity in Fortune 100 and Fortune 500 companies has increased for women and certain minority groups, according to the "Missing Pieces Report: The 2016 Board Diversity Census of Women and Minorities on Fortune 500 Boards," a multiyear study published by the Alliance for Board Diversity (ABD), collaborating with Deloitte for the 2016 Census. However, minority men and women have experienced only slight gains in the last six years, since the ABD first started tracking Fortune 500 board diversity in 2010, up from 12.8 percent in 2010 to 14.4 percent in 2016. The total number of board seats also has not changed much during this time; there were 5,440 total board seats in the Fortune 500 in 2016, compared to 5,463 in 2010.
Key findings include:
In 2016, fewer than 15 percent of all board seats in the Fortune 500 were held by minorities, including African Americans/blacks, Asian/Pacific Islanders and Hispanics/Latino(a)s.
Some progress has been made for African Americans/blacks in securing/holding Fortune 500 board seats, with the bulk of the African American male increases occurring within the Fortune 100. There has been an increase in the Fortune 500 of African American women board members by 18.4 percent, while the total number of African American male board members in the Fortune 500 had only an increase of 1.0 percent. African Americans/blacks appear to have the highest rate of individuals serving on multiple boards – indicating that companies are going to the same individuals rather than expanding the pool of African American/black candidates for board membership.
Asian/Pacific Islanders have shown continued growth in percentage representation on Fortune 500 boards. However, their starting baseline was small – the overall representation of Asian/Pacific Islanders is only 3.1 percent, representing a total of 167 seats.
Hispanic/Latino men picked up a nominal gain of eight board seats, while there was a loss of two Fortune 500 board seats for Hispanic/Latina women since 2012. Overall, Hispanic/Latino(a) men and women hold 188 total board seats, or 3.5 percent of the total.
"The increase in boardroom diversity over the last four years is a small step in the right direction – but it's equally important to note the larger context in which these gains were made. Almost 70 percent of board seats in the Fortune 500 are still held by Caucasian/white men, and at the current rate of progress, we won't likely see the number of women and minorities increase to ABD's target of 40 percent board representation until the year 2026. This is not acceptable," said Ronald C. Parker, ABD chair and president and CEO of The Executive Leadership Council. "A great deal of work remains for corporate board composition to keep pace with the changing demographics of the country at large."
In addition, the report includes data for the Fortune 100 – showing these companies lead the growth in boardroom diversity, outpacing the Fortune 500 with 35.9 percent of women and minorities holding board seats, compared with 30.8 percent in the Fortune 500. Currently, 65 percent of Fortune 100 boards have greater than 30 percent board diversity, compared to just under 50 percent of Fortune 500 companies in the census. However, since the ABD began conducting its census of Fortune 100 board directors in 2004, collective gains for women and minorities have been minimal. In 2016, women and minorities held 35.9 percent of board seats in the Fortune 100, compared to 30.1 percent in 2010 and 28.8 percent in 2004.
Representation of minorities within the Fortune 500 is smaller than in the Fortune 100, where minorities held nearly 20 percent of board seats. Minority men have gained only 1.0 percent in the period 2004-16, and today 82.5 percent of total Fortune 100 seats are held by Caucasian/white individuals.
While nearly 50 percent of Fortune 500 boards have reached at least 30 percent women and minorities on their boards, the "recycle rate" – or rate at which individuals serve on more than one board – is higher for women and minorities than Caucasian/white men, which indicates that while diversity on boards may be increasing, there needs to be a broader net cast to capture "new" women and minorities on boards. This rate also points to the need to look at a broader set of networks, backgrounds, skills and experiences in potential board candidates.
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HOME › Bookshop › History, Politics & Society › RICHARD DUKE OF GLOUCESTER AS LORD PROTECTOR AND HIGH CONSTABLE OF ENGLAND
Richard Duke of Gloucester as Lord Protector and High Constable of England
by Annette Carson
Acclaimed author Annette Carson continues her Richard III studies by concentrating on his brief protectorate while he was still Duke of Gloucester. Until now traditionally misrepresented, his office as Lord Protector has been poorly understood owing to an over-reliance on foreign reporters ignorant of the precise significance of this very English institution. Of equal importance, and equally disregarded, is his concurrent office as Lord High Constable of England, vested with military and judicial authority on behalf of the crown. This study reviews the development of both offices and their constitutional legitimacy in Gloucester's hands.
Annette Carson was given Honorary Life Membership of the Richard III Society in 2014 for distinguished services to the Society's aims.
She featured as the expert witness on BBC Radio 4's series 'Great Lives' when Richard III was nominated by Philippa Langley, 6 January 2015: now available on podcast and iPlayer.
www.annettecarson.co.uk
Annette Carson
Annette Carson is a freelance writer with a preference for history and biography, and has sold some 50,000 non-fiction books. As Historical Consultant she was a member of the 'Looking For Richard' team that found Richard III in Leicester in 2012. After writing on subjects including aviation (aerobatics) and a biography of rock guitarist Jeff Beck, her first Ricardian book was Richard III: The Maligned King. First published in 2008, it has reached sales of 9,000 and is still regularly updated.
This was followed by Richard III: A Small Guide to the Great Debate. Written as a layman's introduction to the Richard III controversy, it has been described as 'Probably the clearest and shortest and most enjoyable read ever'.
In 2014 she edited, co-authored and co-ordinated publication of Finding Richard III: this is the official account of how the 'Looking For Richard' team independently, and at their own expense, pieced together the historical evidence that indicated the correct location of his grave, and raised the money needed to commission archaeologists to perform the excavation.
Her latest book, published May 2015, reveals new research into the protectorate that preceded Richard III's accession: Richard Duke of Gloucester as Lord Protector and High Constable of England.
These last three are available in print and as Troubador eBooks.
Also by Annette Carson / Related Books
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"For So Long As the World Shall Endure": The 500 Year History of Bond's and Ford's Almshouses
by Bond's and Ford's Almshouse Charity
Comfort, Pleasure and Prestige
by Alan Wilson
I'll Take That One
by William B Green
The Development of Cities in Northern and Central Italy during the Renaissance
by Paul N Balchin
Women and the Making of Poetry in Aristo's "Orlando Furioso"
by Ita Mac Carthy
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by Richard Keeble (ed)
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by Jeff Krotz
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by C. F. Carswell
by Donald McGilvray


ERBIL - At least four people were wounded overnight on Tuesday in two bomb explosions in Kirkuk province, locals reported.
According to security sources, the first attack was carried out via a bomb attached to car, wounding three people in a district in central Kirkuk.
An explosive charge fixed to a car went off in a district in central Kirkuk," the source told Iraqi website Baghdad Today. "The explosion left three people injured," added the source.
"Another booby-trapped vehicle went off near a power plant in Domeez area, in southern Kirkuk, leaving a person dead," the source pointed out, as cited by Iraqi News, adding that the second blast resulted in the injuring of one person.
Violations, including killing or kidnapping people, bomb explosions, and similar terrorist activities have increased after the Kurdish Peshmerga troops were forced out by the Iraqi forces from the disputed territories.


text:
An Autonomous EarthObserving Sensorweb (2005)
by S Chien, B Cichy, A Davies, D Tran, G Rabideau, R Castano, R Sherwood, D Mandl, S Frye, S Shulman, J Jones, S Grosvenor
Venue: IEEE Intelligent Systems
A Multi-agent Space, In-situ Volcano Sensorweb
by Wenzhan Song, Behrooz Shirazi, Richard Lahusen
"... We have deployed and demonstrated operations of an integrated space in-situ sensorweb for monitoring volcanic activity. This sensorweb includes a network of ground sensors deployed to the Mount Saint Helens volcano as well as the Earth Observing One spacecraft. The ground operations and space operat ..."
We have deployed and demonstrated operations of an integrated space in-situ sensorweb for monitoring volcanic activity. This sensorweb includes a network of ground sensors deployed to the Mount Saint Helens volcano as well as the Earth Observing One spacecraft. The ground operations and space operations are interlinked in that ground-based intelligent event detections can cause the space segment to acquire additional data via observation requests and space-based data acquisitions (thermal imagery) can trigger reconfigurations of the ground network to allocate increased bandwidth to areas of the network best situated to observe the activity. The space-based operations are enabled by an automated mission planning and tasking capability which utilizes several Open Geospatial Consortium (OGC) Sensorweb Enablement (SWE) standards which enable acquiring data, alerts, and tasking using web services. The ground-based segment also supports similar protocols to enable seamless tasking and data delivery. The space-based segment also supports onboard development of data products (thermal summary images indicating areas of activity, quicklook context images, and thermal activity alerts). These onboard developed products have reduced data volume (compared to the complete images) which enables them to be transmitted to the ground more rapidly in engineering channels.. 1
Organizing Commitee
by Luis Castillo Vidal, Steve Chien, Riccardo Rasconi, Luis Castillo Vidal, Steve Chien, Riccardo Rasconi, Luis Castillo Iactive, Riccardo Rasconi , 2013
"... Proceedings of the 7 th Scheduling and Planning Applications woRKshop ..."
Proceedings of the 7 th Scheduling and Planning Applications woRKshop
Supervisors:
by Jürgen Leitner, Tapio Taipalus , 2009
"... Firstly I want to thank everybody involved in the SpaceMaster programme and everybody I met during these unforgettable last two years. Especially I want to thank my supervisor, Professor Aarne Halme, who allowed me to pursue a research in cooperation with the Intelligent Space Systems Laboratory, at ..."
Firstly I want to thank everybody involved in the SpaceMaster programme and everybody I met during these unforgettable last two years. Especially I want to thank my supervisor, Professor Aarne Halme, who allowed me to pursue a research in cooperation with the Intelligent Space Systems Laboratory, at the University of Tokyo. I want to thank Professor Shinichi Nakasuka for providing me with the possibility to work and do research at his lab in Tokyo for 3 months, as well as helping me with the various problems arising during my thesis. It was an amazing experience and I will never forget it! I also want to thank Professor Keiki Takadama and his students for surviving a very long presentation about me and my thesis at the University of Electro-Communications in Chofu and for helping me, especially with the machine learning part of the thesis. I would also like to thank my instructor Tapio Taipalus and Marek Matusiak for their guidance in selecting, researching, and implementing this thesis. Also thanks to Professor Kalevi Hyyppä for feedback on early thesis drafts.
1 Towards an Autonomous Space In-situ Marine Sensorweb
by S. Chien, J. Doubleday, D. Tran, D. Thompson, G. Mahoney, Y. Chao, R. Castano, J. Ryan, R. Kudela, S. Palacios, D. Foley, A. Balasuriya, H. Schmidt, O. Schofield, M. Arrott, M. Meisinger, S. Frye, L. Ong
"... and ..."
LIGHTS OUT AUTONOMOUS OPERATION OF AN EARTH OBSERVING SENSORWEB
"... The Earth Observing-1 Mission has been operating an autonomous, integrated sensorweb linking dozens of sensor nodes in 24/7 operations since 2004. Key to the successful lights out automation has been flexible web-based tasking with back-end mission operations automation. This sensorweb has achieved ..."
The Earth Observing-1 Mission has been operating an autonomous, integrated sensorweb linking dozens of sensor nodes in 24/7 operations since 2004. Key to the successful lights out automation has been flexible web-based tasking with back-end mission operations automation. This sensorweb has achieved a number of impacts including: 1. Routine re-tasking based on sensor alerts and scientistdefined campaigns with no additional operational effort. 2. Rapid response within hours to changing requests based on weather, science phenomena, and operational concerns. 3. Integration of triggers from the EO-1 mission, other space-borne assets, ground-based in situ instrumentation, and composite synthetic sensors (automatically generated updates such as aviation advisories). 4. Dramatic reduction in operations costs (over $1M US per year). We describe our extensible sensorweb operations architecture, engineering and project results from operational experiences, and our ongoing effort to develop a web service interface to task, retrieve, and process science data. 1.
...to extract the volcanic eruption's signature, downlinking this vital information within hours. The EO-1 sensorweb has demonstrated this and similar scenarios since its first operations in August 2003 =-=[2]-=-. A wide range of operations satellite/platforms make their data freely available (e.g. broadcast or internet) in a rapid fashion (tens of minutes to several hours from acquisition). For example, data...
"... This report represents the extensive effort and support of many individuals both within and outside of the National Aeronautics and Space Administration (NASA) and from the sensor web community. The members of the Earth Science Technology Office (ESTO) Advanced Information Systems Technology (AIST) ..."
This report represents the extensive effort and support of many individuals both within and outside of the National Aeronautics and Space Administration (NASA) and from the sensor web community. The members of the Earth Science Technology Office (ESTO) Advanced Information Systems Technology (AIST) projects who attended the meeting are listed in the breakout group reports (Sections 3.1, 4.1 and 5.1). The editors would like to acknowledge the work of the AIST team lead, Karen Moe, and the primary authors, Bradley Hartman, Samuel Gasster, and Peter Eggan, from The Aerospace Corporation. The editors would also like to acknowledge the support from Sheri Benator, The Aerospace Corporation, for discussions and suggestions regarding architecture frameworks for sensor webs and Penny Newsome, Global Sciences and
Signed: Name: Date:
by Deshendran Moodley , 2009
"... As the candidate's supervisor I have approved this dissertation for submission. ..."
As the candidate's supervisor I have approved this dissertation for submission.
...itoring [123] and visual scene surveillance [17]; for observing natural phenomena such as observing coastlines [41]; weather forecasting [22], measuring snowpack [56] and monitoring volcano eruptions =-=[47]-=-. The resultant increase in sensor data has triggered research activity into the development of infrastructure support services to collect, aggregate and process sensor data for diverse monitoring app...
Towards an Autonomous Space In-situ Marine Sensorweb
"... We describe ongoing efforts to integrate and coordinate space and marine assets to enable autonomous response to dynamic ocean phenomena such as algal blooms, eddies, and currents. Thus far we have focused on the use of remote sensing assets (e.g. satellites) but future plans include expansions to u ..."
We describe ongoing efforts to integrate and coordinate space and marine assets to enable autonomous response to dynamic ocean phenomena such as algal blooms, eddies, and currents. Thus far we have focused on the use of remote sensing assets (e.g. satellites) but future plans include expansions to use a range of in-situ sensors such as gliders, autonomous underwater vehicles, and buoys/moorings. T I.
Abstract — The autonomous Model-based Volcano Sensor
by Ashley Gerard Davies, Daniel Q. Tran, Lukas M, Kate Boudreau, Johanna Cecava, Philip Kyle, Robert Wright
"... Web (MSW), based at JPL, proved its worth during a volcanic ..."
Web (MSW), based at JPL, proved its worth during a volcanic
"... mate goals:"Our job is to reclaim America for Christ, whatever the cost. " Kennedy said at a February 2005 conference,"As the vice regents of God, we are to exercise godly dominion and influence over our neighborhoods, our schools, our government, our literature and arts, our sports arenas, our ente ..."
mate goals:"Our job is to reclaim America for Christ, whatever the cost. " Kennedy said at a February 2005 conference,"As the vice regents of God, we are to exercise godly dominion and influence over our neighborhoods, our schools, our government, our literature and arts, our sports arenas, our entertainment media, our news media, our scientific endeavors—in short, over every aspect and institution of human society." Assuming you have become more concerned, here are some suggestions. Engage in discussions with fellow citizens who may be less enlightened or confused about why President Bush's comments are of such concern or why Judge Jones ruled against the Dover Area School Board. Present them with the following argument: Suppose ID is indeed taught in science classrooms as an alternative explanation of all natural phenomena and that there is an ID counter to evolution, to plate tectonics, to the age of the Earth and its immense history of deep time, to the Big Bang, and even to Hurricane Katrina, and so forth.What will these explanations consist of? In the context of the scientific method, which surely (?) would remain taught in science classrooms, what testable hypotheses can be formulated and tested by gathering what facts/observations to demonstrate ID as a viable "different school of thought"?


When the unit is manufactured, the ID is already set.
receiver, an ID should be set again.
5. Press PAIRING button on the wireless receiver.
receiver). This ferrite core can reduce noises.
product nearby, interference may occur.
Wireless Receiver within 10 m (32 ft.).
occur if longer distance is used.
each Setup menu item, see pages 26 to 30.
details about the resolution (page 22).
Outputs 1 080 lines of progressive video.
Outputs 1 080 lines of interlaced video.
Rear speaker and Front speakers.
and purchase videos using the DivX® VOD service.
Angle – S elected angle/ total number of angles.
The VIRTUAL is VSM Plus.
it to USB (Refer to the page 34).
Preset some radio stations, see page 37 for details.
DivX Certified® device that plays DivX video.


Succulents - Wedding Placecards – Newport Street Paper Co.
Our Succulents folded placecards are designed to complement the full Succulents wedding stationery range.
Each placecard measures 82 x 38mm when folded, and is printed on to 300gsm white Fresco Gesso card, which is lightly textured for a tactile finish. The dark green background will really stand out on your wedding tables, and the original succulent illustration continues on to the reverse of each card.
Placecards come pre-cut and folded, and are fully personalised with either a first name or combined first name and surname. Coordinating On the Day items are also available, be sure to check out our shop for matching items.
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Please note colours may appear differently on different screens.
To be sure of the finish, why not order an Invitation Sample set?


text:
\section{Introduction}
Clifford (super)algebras or special Grassmann (super)algebras play an important role in many branches of
mathematics such as Clifford analysis, algebras, mathematics physics, geometry and topology etc., see for
example \cite{abs,V}.  For the finite dimensional superalgebras, there is a natural graded Morita theory,
which states that a finite dimensional superalgebra is graded Morita  equivalent to a
 graded basic superalgebra, see \cite[Theorem 2.3]{han} for a more general context.

     Note that some important algebraic objects such as
   the Hochschild and cyclic (co)homology of superalgebras are graded Morita invariants \cite{zhao} and
   that the fundamental relationship between Clifford (super)algebras and Bott periodicity is established by
 Atiyah, Bott and Shapiro \cite{abs}, see also \cite{Lawson} for a exposition. It is a natural and interesting problem
   to determine explicitly the graded basic superalgebras for Clifford superalgebras. In this note, using a
    variation of the graded Morita equivalent theory for finite dimensional
superalgebras (Proposition-Definition~\ref{psbasicdef}), we determine explicitly the graded basic superalgebras for
all real and complex Clifford superalgebras (Theorems~\ref{tclifford}
 and \ref{thm-complex}). As an application, we determine explicitly the Grothendieck groups of $\mathbb{Z}_2$-graded modules over all real and complex Clifford superalgebras (Theorem~\ref{thm: Grothendieck}), which is useful to our
  understanding of the Atiyah-Bott-Shapiro isomorphisms.

The lay-out of this note as follows. We begin in Section~\ref{sec: Preliminaries} with preliminaries on superalgebras
 and fix our notations. In section~\ref{sec: Morita} we review briefly the graded Morita theory for superalgebras
 and give some useful lemmas. The graded basic superalgebras for all real and complex Clifford
  superalgebras are determine explicitly in section~\ref{sec: Main}. Finally, as an application, we determine
   explicitly the Grothendieck groups of $\mathbb{Z}_2$-graded modules
   over all real and complex Clifford superalgebras in section~\ref{application}.

\section{Preliminaries on superalgebras}\label{sec: Preliminaries}
 In this section, we recall some facts on superalgebras and fix the notations, our references are
  \cite{bass}, \cite[Chapter 3]{manin} and \cite[\S 1]{kassel}.

 \begin{point}{}*
Let $K$ be a field and $\mathbb{Z}_2:=\mathbb{Z}/2\mathbb{Z}=\{0,
1\}$. By a \textit{superspace} we mean a $\mathbb{Z}_2$-graded
$K$-vector space $V$, namely a $K$-vector space with a decomposition
into two subspaces $V = V_{0}\oplus V_{1}$. A nonzero element $v$ of
$V_i$ will be called \textit{homogeneous} and we denote its degree by
$|v|=i\in \mathbb{Z}_2$. We will view  $K$ as a superspace concentrated in degree 0.

Given   superspaces $V$ and $W$, we view the direct sum $V\oplus W$
and the tensor product $V\otimes_K W$  as superspaces with $(V\oplus
W)_i = V_i\oplus W_i$, and $(V\otimes_K W)_i= V_0\otimes_K W_i\oplus
V_1\otimes_K W_{1-i}$  for $i\in\mathbb{Z}_2$. With this grading,
$V\otimes_K W$ is called the \textit{skew tensor product} of $V$ and
$W$ and is denoted by $V\widehat{\otimes}_KW$. Also, we make the
vector space $\Hom_K(V, W)$ of all $K$-linear maps from $V$ to $W$
into a superspace by setting that $\Hom_K(V, W)_i$ consists of all
the $K$-linear maps $f: V \rightarrow W$ with $f(V_j)\subseteq
W_{i+j}$ for $i, j\in \mathbb{Z}_2$. Elements of
$\Hom_K(V, W)_0$ (resp.~$\Hom_K(V, W)_1$) will be referred to as
\textit{even (resp.~odd) linear maps}.\end{point}

\begin{point}{}* Recall that a \textit{superalgebra} $A$ is both a superspace
and  an associative  $K$-algebra  with identity such that
$A_iA_j\subseteq A_{i+j}$ for $i, j \in \mathbb{Z}_2$. By a
superalgebra \textit{homomorphism} we mean an even linear map which is an
algebra homomorphism in the usual sense and write $A\cong B$ provided that the superalgebras
$A$ and $B$ are isomorphic.

Given  two  superalgebras $A$ and $B$, the \textit{skew tensor
product} $A\widehat{\otimes}_{K} B$  is again a superalgebra with
the inducing grading and multiplication given by
\begin{align*}(a_1 {\otimes} b_1)(a_2 {\otimes} b_2) =
(-1)^{|b_1||a_2|}a_1a_2{\otimes} b_1b_2, \text{ for } a_i\in A \text{ and } b_i\in B.\end{align*}
Note this and other such expressions only make sense for
homogeneous elements.
Observe that the skew tensor product $A\widehat{\otimes}_{K} A\widehat{\otimes}_{K}\cdots\widehat{\otimes}_{K}A$ ($n$ factors) is well-defined and that the \textit{supertwist
map} $T_{A,B}: A\widehat{\otimes}_K B\rightarrow
B\widehat{\otimes}_K A$, $a\otimes b \mapsto (-1)^{|a||b|}b\otimes
a$, for  $a\in A, b \in B$, is an isomorphism of superalgebras. \end{point}

\begin{point}{}*The main principle for superalgebras is the following  \textit{rule
of signs} \cite[\S III.4]{good} or \cite[\S3.1.1]{manin}: if in some formula of usual algebra there are
monomials with interchanged terms, then in the corresponding formula
in superalgebra every interchange neighboring terms, say $a$ and
$b$, is accompanied by the multiplication of the monomial by the
factor $(-)^{|a||b|}$; or equivalently, each letter $a_i$ appearing
as an argument on the left-hand side (LHS) of the defining equation
has a degree associated with it, and a factor of (-1) is introduced
on the right-hand side (RHS) each time a pair of letters on the LHS,
both of odd degree, appearing in reverse order on the RHS. In order
for this rule make sense it is essential that every letter on the
LHS should appear exactly once on the RHS.
\end{point}

\begin{point}{}*
 Let $A$ be a superalgebra. By  a
graded $A$-module $M$ we mean a $\mathbb{Z}_2$-graded left
$A$-module, that is, $M$ is both a superspace $M = M_0\oplus  M_1$
and a left $A$-module such that $A_iM_j \subseteq M_{i+j}$ for
$i, j \in \mathbb{Z}_2$. We denote by $\widehat{A}$ the superalgebra with the same underlying
superspace as $A$ but new multiplication $\hat{a}\hat{b}=(-1)^{|a||b|}\widehat{ab}$.
If $M$ is a graded $A$-module, let $\widehat{M}$ denote the
$\widehat{A}$-module with $M$ as the underlying superspace and
operators defined as $\hat{a}\hat{m}\!=\!(-1)^{|a||m|}\widehat{am}$ for
$a\in\!A,m\in\!M$.

 Unless  otherwise explicitly stated, all our
modules will be $\mathbb{Z}_2$-graded left modules. We denote by
$\mathrm{Mod} A$ the category of all $A$-modules with morphisms
\begin{align*}\Hom_{\Mod A}(M, N):=\Hom_{\Mod A}(M,
N)_0+\Hom_{\Mod A}(M,N)_1,\end{align*}
where $\Hom_{\Mod A}(M, N)_i$, $i\in\mathbb{Z}_2$, is consisting of
all $K$-linear maps $f$ from $M$ to $N$
 such that $f(M_j)\subseteq N_{i+j}$ and $f(am)=(-1)^{i|a|}af(m)$,
 for all $a\in A$, $m\in M$ and  $j\in\mathbb{Z}_2$. The elements of $\Hom_{\Mod A}(M,
N)_0$ (resp.~$\Hom_{\Mod A}(M, N)_1$)  are called \textit{even (resp.~odd) homomorphisms} form $M$ to $N$. Let $\mathrm{Gr} A$ be the category of all $A$-modules with even
homomorphisms. \end{point}

\section{Graded Morita equivalent theory}\label{sec: Morita}
\begin{definition}\label{shifts}
Let $A$ be a superalgebra. The \textit{ parity change} (reps.~\textit{suspension})  functor $\pi$ (resp.~$\sigma$) from $\mathrm{Gr} A$ to itself is defined as following:  for
$M=M_0\oplus M_1$ in $\mathrm{Gr} A$, we define the graded
$A$-module $\pi(M)$ (resp.~$\sigma(M)$) by the conditions:
\begin{enumerate}
\item $\pi(M)$ and $\sigma(M)$ are superspaces with $\pi(M)_i=\sigma(M)_i=M_{i+1}$ for $i\in \mathbb{Z}_2$;

\item the module structure on $\pi(M)$ (resp.~$\sigma(M)$) is defined by $a\pi(m):=(-1)^{|a|}\pi(am)$ (resp. $a\sigma(m):=\sigma(am)$)
for homogeneous elements $a\in A$ and $m\in M$.
\end{enumerate}
\vspace{-2.3mm}
For a homomorphism $f$, we define $\pi(f)$ (resp.~$\sigma(f)$) is the
same underlying linear map as $f$.
\end{definition}

\begin{remark} It follows by definition that $\pi^2=\mathrm{Id}_{\mathrm{Gr} A}$ and $\sigma^2=
\mathrm{Id}_{\mathrm{Gr}A}$, which means that the parity change functor $\pi$ and the suspension functor
$\sigma$ are shifts of the category $\mathrm{Gr}A$ in the sense of \cite[Definition 3.2]{zhang} and \cite[\S2]{s}.  \end{remark}

From now on we assume that $S$ is either the parity change functor $\pi$ or the suspension functor
$\sigma$, unless otherwise explicitly stated. For  an  $A$-module $M$, following \cite[\S2]{s}, we
define the $S$-{\it twisted endomorphism superalgebra} of $M$  to be
the superalgebra
\begin{align*}\mathrm{End}_A^{S}(M):=\Hom_{\Gr A}(M,
M)\oplus\Hom_{\Gr A}(S(M), M),\end{align*}
and define the $S$-{\it twisted $\Hom$ functor} to be the
functor\begin{align*}\Hom^{S}_A(M, -): \Gr A\rightarrow \Gr
\End_A^{S}(M),\quad  N\mapsto \Hom_{\Gr A}(M,
N)\oplus\Hom_{\Gr A}(S(M), N).\end{align*}
Denote by $\Mod^{S} A$ the category of all graded $A$-modules
with homomorphisms $\Hom_A^{S}(M, N)$.

\begin{lemma}[cf.~\cite{bass}, p.\!~123] Assume that $A$ is a finite dimensional superalgebras and that
$M$ and $N$ are $A$-modules. Then $\mathrm{Hom}_{\Mod^{\pi} A}(M, N)\!=\!\Hom_{\Mod^{\sigma}
\widehat{A}}(\widehat{M}, \widehat{N})$ and $\Mod^{\pi} A=\Mod A=\Mod^{\sigma}\widehat{A}$.\label{lbass}\end{lemma}

\begin{proof}First note that for $A$-modules $M$ and $N$, by definition,
\begin{align*}&\mathrm{Hom}_{\Mod^{\pi} A}(M, N)=\Hom_{A}^{\pi}(M,N)=\Hom_{\Gr A}(M,N)\oplus\Hom_{\Gr A}(\pi(M),N) \text{ and}\\
&\Hom_{\Mod^{\sigma}
\widehat{A}}(\widehat{M}, \widehat{N})=\Hom_{\widehat{A}}^{\sigma}(\widehat{M},\widehat{N})=\Hom_{\Gr \widehat{A}}(\widehat{M},\widehat{N})\oplus\Hom_{\Gr \widehat{A}}(\sigma(\widehat{M}),\widehat{N}). \end{align*}
Secondly by definition, we have
\begin{align*}\Hom_{\Gr \widehat{A}}(\widehat{M},\widehat{N})&=\{f: \widehat{M}\rightarrow\widehat{N}\mid f(\widehat{M}_i)\subseteq\widehat{N}_i \text{ and } f(\hat{a}\hat{m})=\hat{a}f(\hat{m}), \forall  i\in\mathbb{Z}_2, a\in A, m\in M\!\}\\
&=\{f: M\rightarrow N\mid f(M_i)\subseteq N_i \text{ and }f(am)=af(m), \forall  i\in\mathbb{Z}_2, a\in A, m\in M\!\}\\
&=\Hom_{\Gr A}(M,N);\\
\Hom_{\Gr \widehat{A}}(\sigma(\widehat{M}),\widehat{N})&=\!\!\{\!f\!: \sigma(\widehat{M})\!\rightarrow\!\widehat{N}\!\mid\! f(\sigma(\widehat{M})_i)\!\subseteq\!\widehat{N}_i\text{ and } f(\hat{a}\sigma(\hat{m}))\!\!=\!\!\hat{a}f(\sigma(\hat{m})), \forall  i\!\in\!\mathbb{Z}_2,a\!\in\! A, m\!\in\! M\!\}\\
&=\!\!\{\!f\!: M\!\rightarrow\! N\!\mid\! f(M_i)\!\subseteq\! N_{1+i}\text{ and }f(am)\!=\!(-1)^{|a|}af(m), \forall  i\!\in\!\mathbb{Z}_2,a\!\in\! A, m\!\in\! M\}\\
&=\Hom_{\Gr A}(\pi(M),N).
\end{align*}
Therefore  $\mathrm{Hom}_{\Mod^{\pi} A}(M, N)\!=\!\Hom_{\Mod^{\sigma}
\widehat{A}}(\widehat{M}, \widehat{N})$.

Finally note that the objects of the categories $\Mod^{\pi} A$, $\Mod A$ and $\Mod^{\sigma}\widehat{A}$ are same.
By the first part of the lemma, we only need to show that $\Hom_{\Mod^{\pi} A}(M,N)=\Hom_{\Mod A}(M,N)$ for all $A$-modules $M$ and $N$; furthermore, it suffices to show that $\Hom_{\Gr A}(\pi(M),N)=\Hom_{\Mod A}(M,N)_{1}$ for all $A$-modules $M$ and $N$. Indeed for all $A$-modules $M$ and $N$, by definition,
\begin{align*}\Hom_{\Gr A}(\!\pi(M),N\!)&=\!\{\!f\!:\pi(\!M\!)\!\rightarrow\! N\!\mid\!f(\!\pi(M)_i\!)\!\subseteq\!N_i \text{ and } f(a\pi(m))\!=\!af(\pi(m)), \forall i\!\in\! \mathbb{Z}_2, a\!\in\! A, m\!\in\! M\!\}\\
&=\!\{\!f\!:M\!\rightarrow\! N\!\mid\!f(\!M_{i+1}\!)\!\subseteq\!N_{i} \text{ and } f(\!\pi(am)\!)\!=\!(\!-1\!)^{|a|}af(\!\pi(m)\!), \forall i\!\in\! \mathbb{Z}_2, a\!\in\! A, m\!\in\! M\!\}\\
&=\!\{\!f\!:M\!\rightarrow\! N\!\mid\!f(M_i)\!\subseteq\!N_{1+i} \text{ and } f(am)\!=\!(-1)^{|a|}af(m), \forall i\!\in\! \mathbb{Z}_2, a\!\in\! A, m\!\in\! M\!\}\\
&=\Hom_{\Mod A}(M,N)_{1},\end{align*}
where the second and third equalities follow by Definition~\ref{shifts} that $\pi(M)_i=M_{i+1}$, $a\pi(m)=(-1)^{|a|}\pi(am)$, and that $f(\pi(am)=f(am)$ for homogeneous elements $a\in A$, $m\in M$ and homogeneous homomorphisms $f$.
As a consequence, we complete the proof.
\end{proof}

 We say that a non-zero $A$ module $M$ is \textit{gr-indecomposable} if it is not the direct sum of two
non-zero   modules that  $M$ is
\textit{$S$-indecomposable} if it is not the direct sum of two
non-zero $A$-modules in the category $\mathrm{Mod}^{S} A$.  A
superalgebra is called to be \textit{gr-divisional} (resp.~\textit{gr-local}) if
every nonzero homogeneous element of it is invertible (resp.~either
invertible or nilpotent).

\begin{lemma}[\cite{han}, \S2.2] Let $A$ be a finite dimensional
superalgebra over $K$ and $M\in\Gr A$. Then
\begin{enumerate}\item $M$ is $S$-indecomposable if and only if $M$ is gr-indecomposable.

\item $M$ is gr-indecomposable if and only if
 $\mathrm{End}^S_{A}(M)$ is gr-local.

\item If $M$ is  gr-simple  then
$\mathrm{End}^S_{A}(M)$ is either a gr-divisional superalgebra
concentrated in degree zero, or a gr-divisional superalgebra containing an
odd involuting element.
\end{enumerate}
\end{lemma}

\begin{definition} Let $A$ and $B$ be two finite dimensional
superalgebras over $K$. We say that $A$ and $B$ are $S$-\textit{graded equivalent}, denote
by $A\thickapprox_{S}B$, if the categories $\Gr A$  and $\Gr B$ are equivalent and $B\!\cong\!\End_A^{S}(P)$
for some finitely generated projective $A$-module $P$. \end{definition}

The relationship between $\pi$-graded equivalence and $\sigma$-graded equivalence is the following:
\begin{lemma}\label{rsp}Assume that finite dimensional superalgebras $A$ and $B$ are $\sigma$-graded equivalent.
Then $A$ and $\widehat{B}$ are $\pi$-graded equivalent.
\end{lemma}

\begin{proof}Assume that the categories $\Gr A$ and $\Gr B$ are equivalent and that there is a finitely
 generated projective $A$-module $P$ such that $B\cong \End_A^{\sigma}(P)$. Observe that
 superalgebras $B$ and $\widehat{\widehat{B}}$ are isomorphic and that $\Gr B$ and
 $\Gr \widehat{B}$ are equivalent.  Applying Lemma~\ref{lbass},
  $\End_{\widehat{A}}^{\pi}(\widehat{P})=\End_{\widehat{\widehat{A}}}^{\sigma}(\widehat{\widehat{P}})=
 \End_A^{\sigma}(P)$. The proof is completed.
\end{proof}

Let $A$ be a finite dimensional superalgebra. Then $A$ has a complete set of primitive orthogonal
idempotents and denoted it by  $\{f_1, \cdots, f_n\}$. Let $P_i=Af_i$ be the projective $A$-module corresponding to the primitive orthogonal idempotent $f_i$ of $A$. Then $\widehat{P}_i=\widehat{A}f_i$ and Lemma~\ref{lbass} and \cite[Corollary 3.10]{Da} imply that $\mathrm{Hom}_{\Mod^\pi A}(P_i,
P_j)=\Hom_{\Mod^\sigma A} (\widehat{P}_i,
\widehat{P}_j)=f_i\widehat{A}f_j $  and $\Hom_{\Mod^\sigma A} (P_i,
P_j)=f_iAf_j$.

We say
that two idempotents $f$ and $g$ of $A$ are $S$-\textit{equivalent}
if the $A$-modules $Af$ and $Ag$ are isomorphic in the category $\Mod^S A$. The following is a criterion of $S$-equivalent of idempotents.

\begin{lemma}\label{lequivalent}The idempotents  $f$ and $g$ are
$S$-equivalent if and only if
 there exist homogeneous elements $x\in fAg$ and $y\in gAf$ satisfying $xgy=f$ and
 $yfx=g$.
\end{lemma}
\begin{proof}Note that $\Hom_{\Mod^{S}A}(Af, Ag)$ is either $fAg$ or $f\widehat{A}g$ and that $\Hom_{\Mod^{S}A}
(Ag, Af)$ is either $gAf$ or $g\widehat{A}f$. Assume that $Af$ and $Ag$ are isomorphic in the category $\Mod^{S}A$.
Then there exists invertible
homogenous homomorphisms induced by homogeneous elements $x\in fAg$ and
$y\in gAf$ satisfying $xgy=f$ and  $yfx=g$.
Conversely, suppose that there exist homogeneous elements $x\in fAg$ and $y\in gAf$ satisfying $xgy=f$ and
 $yfx=g$. Then $x$ and $y$ induce natural homomorphisms $Af\rightarrow Ag$, $f\mapsto xg$ and $Ag\rightarrow Af$, $g\mapsto yf$ respectively, which give the desired isomorphisms.
\end{proof}

The following definition is our investigation for all Clifford superalgebras in this note.
\begin{prop-def}[cf.~Proposition 2.4, \cite{han}] Let $A$ be a finite dimensional superalgebra and $J(A)$
the graded Jacobson radical of $A$. We say that the superalgebra $A$ is \textbf{$\mathbf{S}$-graded basic}
if it satisfies the following equivalent conditions:

\begin{enumerate} \item $f_i$  are pairwise non-$S$-equivalent for $1\le
i\le n$.

\item  Any decomposition of $A$ into indecomposable
projective $A$-modules $A = \oplus^n_{i=1}P_i$ satisfies $P_i$ and $P_j$ are not isomorphic in $\Mod^{S}A$ for all $1\le i \neq j\le n$.

 \item $A/J(A)$ is a direct sum of gr-divisional superalgebras.
 \end{enumerate}
\label{psbasicdef}\end{prop-def}

\begin{remark}By \cite[Theorem 2.3]{han}, a finite dimensional superalgebra $A$
 is $S$-graded equivalent to an $S$-graded basic superalgebra. More precisely, let
 $\{e_i\mid 1\leq i\leq n\}$ be a complete set of non-$S$-equivalent orthogonal
 primitive idempotents of $A$. It follows that $A$ is $S$-graded equivalent to
 $\End^S_A(\oplus_{i=1}^nAe_i)$, which is $S$-graded basic.
  \end{remark}

 The following Lemma is the key to the graded Morita classification for Clifford superalgebras.
\begin{lemma}\label{pskewtensor}
Suppose that the superalgebras $A$ and $A'$ are $S$-graded
equivalent to  $B$ and $B'$ respectively. Then $A\widehat{\otimes}_K
A'$ is $S$-graded equivalent to $B\widehat{\otimes}_K B'$.
\end{lemma}

\begin{proof}Without loss of generality, we may assume that both $B$ and $B'$ are $S$-graded basic and that
$\{f_1, \dots, f_r\}$ and $\{f'_1, \dots,
f'_{r'}\}$ are the complete sets of orthogonal primitive idempotents of $B$ and $B'$ respectively.
Now let $\{e_1, \dots, e_n\}$ and $\{e'_1, \dots,
e'_{n'}\}$ be the complete sets of orthogonal primitive idempotents of
$A$ and $A'$ respectively. Then  $\{f_1, \dots, f_r\}\subseteq\{e_1, \dots, e_n\}$ and $\{f'_1, \dots,
f'_{r'}\}\subseteq\{e'_1, \dots,e'_{n'}\}$. Observe that
$\{e_i\otimes e'_j| 1\le i\le n, 1\le j\le n' \}$ and
$\{f_i\otimes f'_j|1\le i\le r, 1\le j\le r'\}$ is a set of
orthogonal idempotents of $A\widehat{\otimes}_K A'$ such that $\sum_{i,j}e_i\otimes e'_j=1_{A}\otimes 1_{A'}$, and
$\{f_i\otimes f'_j|1\le i\le r, 1\le j\le r'\}$ is a set of
orthogonal idempotents of $B\widehat{\otimes}_K B'$ such that $\sum_{i,j}f_i\otimes f'_j=1_{B}\otimes 1_{B'}$.
Note that $\{f_i\otimes f'_j|1\le i\le r, 1\le j\le r'\}\subseteq \{e_i\otimes e'_j| 1\le i\le n, 1\le j\le n' \}$, which implies that the complete set of non-$S$-equivalent
orthogonal primitive idempotents of $A\widehat{\otimes}_K A'$ can be obtained form
$\{f_i\otimes f'_j|1\le i\le r, 1\le j\le r'\}$ by decomposing these orthogonal idempotents. As a consequence,
$A\widehat{\otimes}_K A'$ is $S$-graded  equivalent to
$B\widehat{\otimes}_K B'$.
\end{proof}

We close this section with some remarks.
 \begin{remark}\label{Rem:graded-basic}\begin{enumerate}\item The $\sigma$-graded equivalence
is exactly the so-called \textit{graded Morita equivalence} defined in \cite{zhang,s} for group-graded
 algebras by ignoring the rule of signs in the case of superalgebras.

 \item A finite dimensional superalgebra is $\sigma$-graded equivalent but not $\pi$-graded equivalent to itself.

 \item The rule of signs implies that for finite dimensional superalgebra
 the $\pi$-graded equivalence is deserving of a better understanding than $\sigma$-graded equivalence, that is, for finite dimensional superalgebra
 $A$ the category $\Mod A$ should be study exhaustively.

\item\label{item-pi-sigma} Lemmas~\ref{rsp} and \ref{lequivalent} show that the $S$-graded basic classification of finite dimensional superalgebras can be determined completely by its graded Morita equivalent classification, that is, by its  $\sigma$-graded equivalent classification.
     \end{enumerate}\end{remark}
\section{Graded Morita equivalent classification of Clifford superalgebras}\label{sec: Main}
 From now on we will write $\mathbb{R}$, $\mathbb{C}$
and $\mathbb{H}$ respectively for the real, complex and quaternion
number-fields and view them as superalgebras over $\mathbb{R}$
concentrated on degree zero respectively and we will say a superalgebra to be graded basic if it is either
$\sigma$-graded basic or $\pi$-graded basic (cf.~Remark~\ref{Rem:graded-basic}(iv)). Note that $\mathbb{R}$, $\mathbb{C}$
and $\mathbb{H}$ are gr-divisional superalgebras; in particular, they are graded basic superalgebras according to Proposition-Definition~\ref{psbasicdef}.

\begin{point}{}*\label{pchevalley} Let $p$, $q$ and $r$ be positive integers.
Following Porteous \cite{porteous95}, we
 denote by $\mathbb{R}^{p,
q, r}$ the real quadratic space $\mathbb{R}^{p+q+r}$ with the
quadratic form $\rho(x)=x_1^2+\cdots+ x_{p}^2-x_{p+1}^2-\cdots-x_{p+q}^2$.
The Clifford superalgebra  $\mathbb{R}_{p, q, r}$  on
$\mathbb{R}^{p, q, r}$ is the real unitary superalgebra generated by
odd generators $e_1, \dots, e_{p+q+r}$ subject to the
following relations:\begin{align*}& e_ie_j+e_je_i=0&& \text{for } 1\leq i\neq j\leq p+q+r;\\
& e_i^2=-e_{j+p}^2=1&&\text{for }1\leq i\leq p \text{ and }1\leq j\leq q;\\
&e_{p+q+k}^2=0&&\text{for } 1\le k\leq r.\end{align*}
Note that the Clifford superalgebra
$\mathbb{R}_{0, 0, r}$ is the real Grassmann superalgebra
$\bigwedge_{\mathbb{R}}(r)$ generated by odd generators $e_{p+q+1}, \dots, e_{p+q+r}$. Observe that the orthogonal primitive idempotent of $\bigwedge_{\mathbb{R}}(r)$ is the unit 1 since $\bigwedge_{\mathbb{R}}(r)/J(\bigwedge_{\mathbb{R}}(r))=\mathbb{R}$, which implies $\bigwedge_{\mathbb{R}}(r)$
are  graded basic  for all positive integer $r\ge 1$ according to Proposition-Definition~\ref{psbasicdef}.

For now on we write $\mathbb{R}_{p, q}$ for the Clifford superalgebra
$\mathbb{R}_{p, q, 0}$. It is well-known that there are superalgebras isomorphism $\mathbb{R}_{p, q, r}
\cong \mathbb{R}_{p, q}\widehat{\otimes}_\mathbb{R}\bigwedge_{\mathbb{R}}(r)$ and $\mathbb{R}_{p,
q}\widehat{\otimes}_{\mathbb{R}} \mathbb{R}_{p', q'}\cong
\mathbb{R}_{p+p', q+q'}$ for all positive integers $p$, $q$, $r$, $p'$ and $q'$, see \cite{chev} or \cite[Proposition~1.6]{abs}.\end{point}

\begin{point}{}* Let $V=\mathbb{R}^{p+q+r}$ and $\alpha: V\rightarrow V$, $v\mapsto -v$. Then $\alpha$ induces a
 automorphism of the (ungraded) Clifford algebra $\mathbf{C}\ell(V,\rho)=T(V)/\langle v\otimes v-\rho(v)|v\in V\rangle$
  where $T(V)$ is the tensor algebra of $V$. Since $\alpha^2=\mathrm{Id}$, there is a decomposition
\begin{align*}\mathbf{C}\ell(V,\rho)=\mathbf{C}\ell^0(V,\rho)\oplus \mathbf{C}\ell^1(V,\rho)\end{align*}
where $\mathbf{C}\ell^i(V,\rho)=\{\phi\in\mathbf{C}\ell(V,\rho)\mid\alpha(\phi)=(-1)^i\phi\}$ are the eigenspaces
of $\alpha$. Clearly, since $\alpha(\phi_1\phi_2)=\alpha(\phi_1)\alpha(\phi_2)$, we have that
\begin{align*}\mathbf{C}\ell^i(V,\rho)\mathbf{C}\ell^j(V,\rho)\subseteq \mathbf{C}\ell^{i+j}(V,\rho)\quad
\text{ for all }i,j\in \mathbb{Z}_2,\end{align*}
that is, $\mathbf{C}\ell(V,\rho)$ is a superalgebra.  It is an observation of Atiyah, Bott and Shapiro \cite{abs}
 that this $\mathbb{Z}_2$-grading plays an important role in the analysis and application of Clifford algebras.
\end{point}

\begin{remark}The two superalgebras $\mathbb{R}_{p, q, r}$ and $\mathbf{C}\ell(V,\rho)$ are naturally isomorphic.
\end{remark}

Denote by $\mathbb{D}_{+}$ (resp.~$\mathbb{D}_{-}$) the real superalgebra generated by odd element $e_{+}$
(resp.~$e_{-}$) subject to the relation $e_{+}^2=1$ (resp.~$e^2_{-}=-1$). For a positive integer $n$, we denote by
$\mathbb{D}_{+}^{n}$ (resp.~$\mathbb{D}_{-}^{n}$) the superalgebra
$\mathbb{D}_{+}\widehat{\otimes}_{\mathbb{R}}\cdots\widehat{\otimes}_{\mathbb{R}}
\mathbb{D}_{+}$  (resp.~$\mathbb{D}_{-}\widehat{\otimes}_{\mathbb{R}}\cdots\widehat{\otimes}_{\mathbb{R}}
\mathbb{D}_{-}$) ($n$ factors) and write  $e_{+}^{i_1\dots i_n}$ (resp.~$e_{-}^{i_1\dots i_n}$) for the homogeneous element
$e_{+}^{i_1}\otimes\dots\otimes
e_{+}^{i_n}\in\mathbb{D}_{+}^{n}$ (resp.~$e_{-}^{i_1}\!\otimes\!\cdots\!\otimes\!
e_{-}^{i_n}\!\in\!\mathbb{D}_{-}^{n}$) where $i_j=0, 1$ for all $1\leq j\leq n$. Note that $\mathbb{D}_{+}^n$
 and $\mathbb{D}_{-}^n$ are exactly $\mathbb{R}_{n,0}$ and $\mathbb{R}_{0,n}$ respectively.

\begin{lemma}\label{dd}If $n=1, 2,3$ then $\mathbb{D}^n_{+}$ and $\mathbb{D}^n_{-}$ are
gr-divisional superalgebras.\end{lemma}
\begin{proof}First, it is trivial that $\mathbb{D}_{\pm}$  are gr-divisional superalgebras. Noticing that the even (resp.~odd) elements of $\mathbb{D}_{+}^2$ are of the form $a\otimes 1+be_{+}
\otimes e_{+}$ (resp.~$a\otimes e_{+}+be_{+}\otimes 1$), where $a$, $b\in \mathbb{R}$, and that for
all $a, b\in \mathbb{R}$, \begin{align*}(a\otimes 1+be_{+}\otimes e_{+})
 (a\otimes 1-be_{+}\otimes e_{+})=(a^2+b^2)\otimes 1=
 (a\otimes e_{+}+be_{+}\otimes 1)(a\otimes e_{+}+be_{+}\otimes
 1).\end{align*}
Thus all non-zero homogeneous elements of $\mathbb{D}_{+}^2$ are invertible, that is,
$\mathbb{D}_{+}^2$ is gr-divisional.

Set $\mathbf{1}:=1\widehat{\otimes}1\widehat{\otimes}1$,
$\mathbf{i}_{+}:=e_{+}^{011}$,
$\mathbf{j}_{+}:=e_{+}^{101}$,
$\mathbf{k}_{+}:=e_{+}^{110}$ and
$\theta_{+}:=e_{+}^{111}$. Then it follows directly that $\theta_{+} \mathbf{i}_{+}+\mathbf{i}_{+}\theta_{+}=0$, $\theta_{+} \mathbf{j}_{+}+\mathbf{j}_{+}\theta_{+}=0$ and $\theta_{+} \mathbf{k}_{+}+\mathbf{k}_{+}\theta_{+}=0$. Furthermore, we have $\mathbb{R}\langle\mathbf{1}, \mathbf{i}_{+},
\mathbf{j}_{+}, \mathbf{k}_{+}\rangle\cong \mathbb{H}$ and
$\mathbb{D}_{+}^3\cong\mathbb{H}\oplus\mathbb{H}\theta_{+}$. Note that $\theta_{+}$ is a odd element of $\mathbb{D}^3_{+}$ satisfying $\theta^2_{+}=-1$ and that $\mathbb{H}$ is a gr-divisional superalgebra concentrated on degree zero.  So $\mathbb{D}^3_{+}$ is  gr-divisional.

 Observe that the even (resp.~odd) elements of $\mathbb{D}_{-}^2$ are of the form $a\otimes 1+be_{-}\otimes e_{-}$ (resp.~$a\otimes e_{-}+be_{-}\otimes 1$), where $a$, $b\in \mathbb{R}$, and that for all $a, b\in \mathbb{R}$,
 \begin{align*}(a\otimes 1+be_{-}\otimes e_{-})
 (a\otimes 1-be_{-}\otimes e_{-})=-(a^2+b^2)\otimes 1=
 (a\otimes e_{-}+be_{-}\otimes 1)(a\otimes e_{-}+be_{-}\otimes
 1).\end{align*}
Thus all non-zero homogeneous elements of $\mathbb{D}_{-}^2$ are invertible, that is
$\mathbb{D}_{-}^2$ is gr-divisional.

Set $\mathbf{i}_{-}:=e_{-}^{011}$,
$\mathbf{j}_{-}:=e_{-}^{101}$,
$\mathbf{k}_{-}:=e_{-}^{110}$, and
$\theta_{-}:=e_{-}^{111}$. Then it follows directly that $\theta_{-} \mathbf{i}_{-}+\mathbf{i}_{-}\theta_{-}=0$, $\theta_{-} \mathbf{j}_{-}+\mathbf{j}_{-}\theta_{-}=0$, and $\theta_{-}\mathbf{k}_{-}+\mathbf{k}_{-}\theta_{-}=0$. Moreover, we have $\mathbb{R}\langle\mathbf{1}, \mathbf{i}_{-},
\mathbf{j}_{-}, \mathbf{k}_{-}\rangle\cong \mathbb{H}$ and
$\mathbb{D}_{-}^3\cong\mathbb{H}\oplus\mathbb{H}\theta_{-}$. Note that $\theta_{-}$ is a odd element of $\mathbb{D}^3_{-}$ satisfying $\theta^2_{-}=1$ and that $\mathbb{H}$ is a gr-divisional superalgebra concentrated on degree zero. Thus $\mathbb{D}^3_{-}$ is gr-divisional.
\end{proof}

\begin{lemma}\label{dc} $\mathbb{D}_{+}\widehat{\otimes}_{\mathbb{R}}\mathbb{D}_{-}$
is graded Morita equivalent to the superalgebra
$\mathbb{R}$.\end{lemma}
\begin{proof}Set $A=\mathbb{D}_{+}\widehat{\otimes}_{\mathbb{R}}\mathbb{D}_{-}$. By Proposition-Definition~\ref{psbasicdef},
it suffices to determine the non-$\sigma$-equivalent idempotents of $A$. Let
$f_{\pm}=\frac{1}{2}(1\otimes 1\pm e_{+}\otimes e_{-})$. Then $f_{\pm}^2=f_{\pm}$, $f_{+}+f_{-}=1\otimes 1$ and $f_{+}f_{-}=0=f_{-}f_{+}$, which implies that
$\{f_{+}, f_{-}\}$ is the complete set of orthogonal primitive
idempotents of $A$ since $\dim_{\mathbb{R}}A_{0}=2$. It is straightforward to show that
\begin{align*}f_{+}Af_{-}=\{r(e_{+}\otimes1-1\otimes e_{-})\mid r\in\mathbb{R}\}\quad\text{and}\quad
f_{-}Af_{+}=\{r(e_{+}\otimes1+1\otimes e_{-})\mid r\in\mathbb{R}\}.
 \end{align*}
Let $x=\frac{1}{2}(e_{+}\otimes1-1\otimes e_{-})$ and $y=\frac{1}{2}(e_{+}\otimes1+1\otimes e_{-})$. Then  $xf_{-}y=f_{+}$ and $yf_{+}x=f_{-}$.
Applying Lemma~\ref{lequivalent},  $f_+$ and $f_-$ are
$\sigma$-equivalent orthogonal primitive idempotents of $A$. As a consequence, Proposition-Definition~\ref{psbasicdef} implies that
$A$ is graded Morita equivalent to
$f_{+}Af_{+}=\mathbb{R}f_{+}\cong \mathbb{R}$.
\end{proof}

\begin{lemma}\label{dddd}$\mathbb{D}_{\pm}^4$ are graded Morita
equivalent to the superalgebra $\mathbb{H}$.
\end{lemma}

\begin{proof}By the proof of Lemma~\ref{dd}, the natural graded map $e_{-}\mapsto \theta_{+}$ gives an isomorphism between $\mathbb{D}_{-}{\widehat{\otimes}}_{\mathbb{R}}\mathbb{H}$ and $\mathbb{D}_{+}^3$.
 Therefore, using Lemmas~\ref{pskewtensor} and \ref{dc}, $\mathbb{D}_{+}^4=\mathbb{D}_{+}\widehat{\otimes}_{\mathbb{R}}
 \mathbb{D}_{+}^3\cong \mathbb{D}_{+}\widehat{\otimes}_{\mathbb{R}}\mathbb{D}_{-}\widehat{\otimes}_{\mathbb{R}}\mathbb{H}$
 is graded Morita equivalent to  the superalgebra $\mathbb{H}$. Similarly,
 noticing that $\mathbb{D}_{+}\widehat{\otimes}\mathbb{H}\cong \mathbb{D}_{-}^3$,
 we can show that $\mathbb{D}_{-}^4$ is graded Morita equivalent to the superalgebra $\mathbb{H}$.\end{proof}

\begin{remark}\begin{enumerate}\item The Lemmas~\ref{dc} and \ref{dddd} show that the skew tensor product
of $\sigma$-graded basic superalgebras may not be  $\sigma$-graded basic. In particular,
the skew tensor product of gr-divisional superalgebras may not be gr-divisional.
\item Lemmas~\ref{rsp}, \ref{dc} and \ref{dddd} imply that $\mathbb{D}_{+}\otimes\mathbb{D}_{-}$ and $\mathbb{D}_{\pm}^4$ are also $\pi$-graded equivalent to $\mathbb{R}$ and $\mathbb{H}$ respectively since $\widehat{\mathbb{D}_{\pm}}=\mathbb{D}_{\mp}$.
\end{enumerate}
\end{remark}
Now we can obtain the graded Morita classifications for all real Clifford superalgebras.

\begin{theorem}\label{tclifford} Assume that $p$, $q$ and $r$ are non-negative integers. Then \begin{enumerate}\item
$\mathbb{R}_{p, q, r}\,\approx_{\sigma}\left\{\aligned
&\bigwedge\!_{\mathbb{R}}(r), &&\text{if }p-q\equiv0 (\mathrm{mod\,} 8);\\
&\mathbb{H}\widehat{\otimes}_{\mathbb{R}}\bigwedge\!_{\mathbb{R}}(r),&&\text{if } p-q\equiv 4 (\mathrm{mod\,}8);\\
&\mathbb{D}_{+}^i\widehat{\otimes}_{\mathbb{R}}\bigwedge\!_{\mathbb{R}}(r), &&\text{if }p-q\equiv  4-i(\mathrm{mod\,}8)\text{ for } 1\leq i\leq 3;\\
&\mathbb{D}_-^i\widehat{\otimes}_{\mathbb{R}}\bigwedge\!_{\mathbb{R}}(r),&&\text{if }p-q\equiv 4+i (\mathrm{mod\,}8)
\text{ for } 1\le i \leq 3.
\endaligned\right.$

\item $\mathbb{R}_{p, q, r}\,\approx_{\pi}\left\{\aligned
&\bigwedge\!_{\mathbb{R}}(r), &&\text{if }p-q\equiv0 (\mathrm{mod\,} 8);\\
&\mathbb{H}\widehat{\otimes}_{\mathbb{R}}\bigwedge\!_{\mathbb{R}}(r),&&\text{if } p-q\equiv 4 (\mathrm{mod\,}8);\\
&\mathbb{D}_{+}^i\widehat{\otimes}_{\mathbb{R}}\bigwedge\!_{\mathbb{R}}(r), &&\text{if }p-q\equiv  4+i(\mathrm{mod\,}8)\text{ for } 1\leq i\leq 3;\\
&\mathbb{D}_-^i\widehat{\otimes}_{\mathbb{R}}\bigwedge\!_{\mathbb{R}}(r),&&\text{if }p-q\equiv 4-i (\mathrm{mod\,}8)
\text{ for } 1\le i \leq 3.
\endaligned\right.$
\end{enumerate}
$($Note that these superalgebras are graded basic superalgebras.$)$
\end{theorem}
\begin{proof}(i) Note that $\mathbb{D}_+\cong \mathbb{R}_{1, 0}$ and $\mathbb{D}_{-}\cong \mathbb{R}_{1, 0}$.
Using \cite[Proposition~1.6]{abs}, Lemmas~\ref{pskewtensor} and \ref{dc},
$$\mathbb{R}_{p,q}=\mathbb{D}_{+}^p\widehat{\otimes}_{\mathbb{R}}\mathbb{D}_{-}^q
\,{\stackrel{\sigma}\approx}\left\{\aligned\mathbb{D}^{p-q}_+,&\, \text{\quad\quad if } p\ge q;\\
\mathbb{D}_-^{q-p},&\,\text{\quad\quad otherwise}.\endaligned\right.
$$
\indent Now by \cite[\S 5.6]{porteous95}, $\mathbb{H}\widehat{\otimes}_{\mathbb{R}}\mathbb{H}$ is isomorphic to the $4\times 4$ real matrix superalgebra concentrated
on degree zero, which is (graded) Morita equivalent to  $\mathbb{R}$. Consequently, by Lemma~\ref{dddd}, $\mathbb{D}_{\pm}^8$ is graded Morita equivalent to $\mathbb{R}$.  Therefore Lemmas~\ref{pskewtensor} and
\ref{dddd} imply that
\begin{enumerate}\item if $p\geq q$ then $\mathbb{D}^{p-q}_+\,\approx_{\sigma}\left\{\aligned\mathbb{D}_{+}^i,\quad&\,\text {\quad\quad if } p-q\equiv
i(\text{mod\,}8), i=0, 1, 2, 3,\\\mathbb{D}_{+}^i\widehat{\otimes}_{\mathbb{R}}\mathbb{H},&\,
\text {\quad\quad if } p-q\equiv 4+i(\text{mod\,}8), i=0, 1, 2, 3;
\endaligned\right. $
\vspace{1\jot}
\item if $p<q$ then $\mathbb{D}^{q-p}_-\,\approx_{\sigma}\left\{\aligned\mathbb{D}_{-}^i,\quad&\,\text {\quad\quad if } q-p\equiv
i(\text{mod\,}8), i=0, 1, 2, 3,\\\mathbb{D}_{-}^i\widehat{\otimes}_{\mathbb{R}}\mathbb{H},&\,
\text {\quad\quad if } q-p\equiv 4+i(\mathrm{mod\,}8), i=0, 1, 2, 3;
\endaligned\right. $\end{enumerate}
By the proof of Lemma~\ref{dddd}, $\mathbb{D}_{+}\widehat{\otimes}_{\mathbb{R}}\mathbb{H}\cong
\mathbb{D}_{-}^3$ and $\mathbb{D}_{-}\widehat{\otimes}_{\mathbb{R}}\mathbb{H}\cong
\mathbb{D}_{+}^3$. Applying Lemmas~\ref{pskewtensor} and \ref{dc} again, we have  $\mathbb{D}_{+}^{i}\widehat{\otimes}_{\mathbb{R}}\mathbb{H}\approx_{\sigma}\mathbb{D}_{-}^{4-i}$ and  $\mathbb{D}_{-}^{i}\widehat{\otimes}_{\mathbb{R}}\mathbb{H}\approx_{\sigma}\mathbb{D}_{+}^{4-i}$ for $1\le i\le 3$.
Noticing that $q-p\equiv 4+i (\mathrm{mod\,}8)$ if and only if $p-q\equiv 4-i (\mathrm{mod\,}8)$.
As a consequence we complete the proof of (i) by \cite[Proposition~1.6]{abs}.

Note that $\widehat{\mathbb{R}_{p,q,r}}\cong \mathbb{R}_{q,p,r}$. Hence (ii) is follows directly by Lemma~\ref{rsp} and (i). \end{proof}

\begin{remark}Note that the $\mathbb{Z}_2$-graded Hochschild and cyclic (co)homology $H_*(A)$ of a finite dimensional superalgebra $A$ is graded Morita equivalent invariant \cite{zhao}. Theorem~\ref{tclifford} implies that $$H_{*}(\mathbb{R}_{p,q})=\left\{\aligned
&0,&&\text{if }*\geq 1;\\
&\mathbb{R},&&\text{if } *=0,
\endaligned\right.$$
since $\mathbb{R}$, $\mathbb{H}$ and  $\mathbb{D}_{\pm}^i$ for all $1\leq i\leq 3$ are separable superalgebras (see
\cite[Chap.~IV-V]{bass}, which gives a slight generalization of \cite[Proposition 1]{kassel} since the quadratic form  $\rho(x)=\sum_{i=1}^px_i^2-\sum_{j=1}^qx_{p+j}^2$ on $\mathbb{R}^{p+q}$ is degenerate.
\end{remark}

\begin{point}{}* Let $p$ and $q$ be positive integers. We denote by
$\mathbb{C}^{p, q}$ the  complex quadratic space $\mathbb{C}^{p+q}$
with a quadratic form $x_1^2+\cdots+ x_{p}^2$. The
Clifford superalgebra $\mathbb{C}_{p, q}$  on $\mathbb{C}^{p, q}$ is the complex unitary
superalgebra generated by odd generators $e_1, \cdots, e_{p+q}$ subject to the following relations:
\begin{align*}& e_ie_j + e_je_i = 0 &&\text{for } 1\le i\neq  j\le p+q;\\
& e_i^2=1&&\text{for }1\leq i\leq p;\\
&e_{p+j}^2=0&&\text{for } 1\leq j\leq q.\end{align*}
Note that the Clifford superalgebra $\mathbb{C}_{0, q}$ is the complex
Grassman superalgebra $\bigwedge_{\mathbb{C}}(q)$ generated by odd generators $e_{p+1}$, $\dots$, $e_{p+q}$.
Observe that the orthogonal primitive idempotent of $\bigwedge_{\mathbb{C}}(q)$ is the unity 1,
which implies that $\bigwedge_{\mathbb{C}}(q)$ are graded basic superalgebras for all positive integer $q\ge 1$.

We denote by $D=\mathbb{C}\oplus \mathbb{C}\varepsilon$ the complex
superalgebra generated by odd generators $\varepsilon$ subject to the relation
$\varepsilon^2=1$. For positive integer $n$, let $D^n$ be the superalgebra
$D\widehat{\otimes}_{\mathbb{C}}\cdots\widehat{\otimes}_{\mathbb{C}}
D$ ($n$ factors). Then $D^p\cong \mathbb{C}_{p,0}$ and $\mathbb{C}_{p, q}\cong
D^p\widehat{\otimes}_{\mathbb{C}}\bigwedge_{\mathbb{C}}(q)$  for all positive integers $p$ and $q$, see  \cite[Proposition~1.6]{abs}.\end{point}

\begin{lemma}\label{DD}
$D\widehat{\otimes}_{\mathbb{C}}D$ is graded Morita equivalent to
the superalgebras $\mathbb{C}$.
\end{lemma}

\begin{proof}Set $A=D\widehat{\otimes}_{\mathbb{C}}D$
Let $\varepsilon_{\pm}=\frac{1}{2}(1\otimes 1\pm
\mathbbm{i}\varepsilon\otimes\varepsilon)$, where $\mathbbm{i}$ is the
imaginary unit of $\mathbb{C}$. Then $\varepsilon_{\pm}^2=\varepsilon_{\pm}$, $\varepsilon_{+}\varepsilon_{-}=0=\varepsilon_{-}\varepsilon_{+}$ and $\varepsilon_{+}+\varepsilon_{-}=1\otimes 1$, which implies that  $\{\varepsilon_{+}, \varepsilon_{-}\}$ is the
complete set of orthogonal primitive idempotents of $D^2$ since $\dim_{\mathbb{C}}A_{0}=2$. It is straightforward to show that \begin{align*}\varepsilon_{+}A\varepsilon_{-}=\{c(\mathbbm{i}\otimes\varepsilon-1\otimes \varepsilon)\mid c\in\mathbb{C}\}\quad\text{and}\quad
\varepsilon_{-}A\varepsilon_{+}=\{c(\mathbbm{i}\otimes \varepsilon+\varepsilon\otimes1)\mid c\in\mathbb{C}\}.
 \end{align*}
Let $x=\frac{1}{2}(\mathbbm{i}\otimes\varepsilon-1\otimes \varepsilon)$ and $y=\frac{1}{2}(\varepsilon\otimes1+\mathbbm{i}\otimes \varepsilon)$. Then  $x\varepsilon_{-}y=\varepsilon_{+}$ and $y\varepsilon_{+}x=\varepsilon_{-}$.
Applying Lemma~\ref{lequivalent},  $\varepsilon_+$ and $\varepsilon_-$ are
$\sigma$-equivalent orthogonal primitive idempotents of $A$. Thus Proposition-Definition~\ref{psbasicdef} implies
 that $A$ is graded Morita equivalent to $\varepsilon_{+}A\varepsilon_{+}=\mathbb{C}\varepsilon_{+}\cong\mathbb{C}$.
\end{proof}

Now we yield the graded Morita equivalent classification for all complex Clifford superalgebras.

\begin{theorem}\label{thm-complex}Assume that $p$ and $q$ are non-negative integers. Then $\mathbb{C}_{p,q}$ is graded Morita equivalent to graded basic superalgebra $\mathbb{C}_{p(\mathrm{mod\,}2),q}$.
\end{theorem}

\begin{proof}Note that $\mathbb{C}_{p, q}\cong
\mathbb{C}_{1, 0}\widehat{\otimes}_{\mathbb{C}}\bigwedge_{\mathbb{C}}(q)$ and $\mathbb{C}_{1, 0}\cong D$.
It follows immediately that $\mathbb{C}_{p,q}\approx_{\sigma}\mathbb{C}_{p(\mathrm{mod\,}2),q}$ by using Lemmas~\ref{DD} and \ref{pskewtensor}. On the other hand, note that $\widehat{\mathbb{C}_{p,q}}\cong \mathbb{C}_{p,q}$, we have $\mathbb{C}_{p,q}\approx_{\pi}\mathbb{C}_{p(\mathrm{mod\,}2),q}$.  We complete the proof.
 \end{proof}

\section{The Grothendieck groups of Clifford superalgebras}\label{application}
\begin{point}{}* Most of the important applications of Clifford Superalgebras come through a detailed understanding of their ($\mathbb{Z}_2$-graded) representations. This understanding follows rather easily form the graded Morita equivalent classification given by Theorems~\ref{tclifford} and \ref{thm-complex}. In this section, we shall restrict our attention to the Clifford superalgebras $\mathbb{R}_{p,q}$ and $\mathbb{C}_{p,0}$ for all positive integers $p$ and $q$ in order to simplify our presentation, which are exactly those Clifford (super)algebras play a fundamental role in \cite{abs,Lawson}.\end{point}

\begin{point}{}*We begin with some notations. For positive integers $p$ and $q$, let $\mathrm{Irr}_{p,q}$ be the set of
 all nonequivalent irreducible graded $\mathbb{R}$-module for $\mathbb{R}_{p,q}$ in the category $\Gr \mathbb{R}_{p,q}$.
  Similarly, let $\mathrm{Irr}_{p}^{\mathbb{C}}$ be the set of all nonequivalent irreducible graded $\mathbb{R}$-module
   for $\mathbb{C}_{p,0}$ in the category $\Gr \mathbb{C}_{p,0}$.

An object which will be of our interest is the following.  Let $K_{p,q}$ be the Grothedieck group of
the category of finite dimensional graded real representations of $\mathbb{R}_{p,q}$, and let
$K_{p}^{\mathbb{C}}$ be the Grothedieck group of the category of
finite dimensional graded complex representations of $\mathbb{C}_{p,0}$. Then $K_{p,q}$ and $K_{p}^{\mathbb{C}}$
are the free abelian groups generated by $\mathrm{Irr}_{p,q}$ and $\mathrm{Irr}_{p}^{\mathbb{C}}$ respectively.
\end{point}

From the classification of Theorems~\ref{tclifford} and \ref{thm-complex} we immediately conclude the following:
\begin{lemma}\label{lem-v_p,q}Let $v_{p,q}$ and $v_p^{\mathbb{C}}$ denote the cardinality of $\mathrm{Irr}_{p,q}$
and of $\mathrm{Irr}_{p}^{\mathbb{C}}$ respectively. Then $$v_{p,q}=\left\{\begin{array}{ll}
2, & \text{ if }p-q\equiv 0 (\mathrm{mod\,}4) \\
 1, & \hbox{otherwise}\end{array}\right. \quad \text{ and }\quad
v_{p}^{\mathbb{C}}=\left\{\begin{array}{ll}2, & \text{ if }p \text{ is odd}\\
                                              1, &\text{ if }p \text{ is even}.
                                            \end{array}
                                          \right.$$\end{lemma}
 Now we can obtain the main result of this section.
\begin{theorem}\label{thm: Grothendieck}Let $p$ and $q$ be positive integers.
\begin{enumerate}\item The elements $\mathrm{Irr}_{p,q}$, $v_{p,q}$ and $K_{p,q}$  are as given in the following table:

\vspace{1.5\jot}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
  \hline
   $p-q(\mathrm{mod\,}8)$ & $0$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$&$7$ \\ \hline
  $\mathbb{R}_{p,q}/\!\!\approx$& $\mathbb{R}$ & $\mathbb{R}_{1,0}$ & $\mathbb{R}_{2,0}$ & $\mathbb{R}_{3,0}$ & $\mathbb{H}$ & $\mathbb{R}_{0,1}$ & $\mathbb{R}_{0,2}$ &$\mathbb{R}_{0,3}$\\
$\mathrm{Irr}_{p,q}/\!\!\approx$ & $\{\mathbb{R}, \sigma(\mathbb{R})\}$ & $\{\mathbb{R}_{1,0}\}$ & $\{\mathbb{R}_{2,0}\}$ & $\{\mathbb{R}_{3,0}\}$ &$\{\mathbb{H}, \sigma(\mathbb{H})\}$ & $\{\mathbb{R}_{0,1}\}$ & $\{\mathbb{R}_{0,2}\}$&$\{\mathbb{R}_{0,3}\}$ \\
  $v_{p,q}$ & $2$ & $1$ & $1$ & $1$ & $2$ & $1$ & $1$&$1$ \\
$K_{p,q}$ & $\mathbb{Z}\oplus\mathbb{Z}$ & $\mathbb{Z}$ & $\mathbb{Z}$ & $\mathbb{Z}$ &$\mathbb{Z}\oplus \mathbb{Z}$ & $\mathbb{Z}$ & $\mathbb{Z}$&$\mathbb{Z}$ \\
  \hline
\end{tabular}

\vspace{1\jot}
\item $v_{p}^{\mathbb{C}}$, $\mathrm{Irr}_{p}^{\mathbb{C}}$ and $K_{p}^{\mathbb{C}}$ are given by following table:

\vspace{1.5\jot}\begin{tabular}{|c|c|c|c|c|}
                  \hline
                                   $p(\mathrm{mod\,}2)$ & $\mathbb{C}_p/\!\!\approx$& $\mathrm{Irr}_{p}^{\mathbb{C}}/\!\!\approx$&$v_{p}^{\mathbb{C}}$&$K_{p}^{\mathbb{C}}$ \\ \hline
                  0 & $\mathbb{C}$ & $\{\mathbb{C}, \sigma(\mathbb{C})\}$&$2$&$\mathbb{Z}\oplus\mathbb{Z}$ \\
                  1 & $\mathbb{C}_{1,0}$ & $\{\mathbb{C}_{1,0}\}$&$1$&$\mathbb{Z}$ \\
                  \hline
                \end{tabular}
\end{enumerate}
\end{theorem}

\begin{proof}Note that the graded modules of a gr-divisional superalgebra without non-trivial odd part are some copies of this gr-divisional superalgebra. Then the theorem is a direct consequence of Theorems~\ref{tclifford} and  \ref{thm-complex} and Lemma~\ref{lem-v_p,q}.\end{proof}

We remark in closing that Atiyah, Bott and Shapiro obtained Theorem~\ref{thm: Grothendieck}
 by using the periodicity theorem for Clifford superalgebras \cite[\S~5]{abs}, and it is
 important to understand the Atiyah-Bott-Shapiro isomorphisms \cite[Theorem~11.5]{abs}, see \cite{abs} and \cite{Lawson} for more details.

\section*{Acknowledgements}

The note develops partly form a part of \cite{zhaophd}.
The author is very grateful to Professors Yang Han and Yingbo Zhang for their invaluable help.
The author would like to thank the referees for their comments and suggestions, which contributed to improvements in the presentation of the results.

\bibliographystyle{amsplain}

redpajama_set_name: RedPajamaArXiv


Utrera (Spanish pronunciation: [uˈtɾeɾa]) is a municipality in south-west Spain. It is in the province of Seville, in the autonomous community of Andalusia. As of 2008 it has a population close to 50,202.
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text:
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