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Artur Popławski has been at 1 events

HostFollowersTitleDateGuestsLinks
NASA2,171,054Join us for a live Google+ Hangout on observing the Earth while in orbit and the importance of Earth science observations from space.  The astronauts living aboard the International Space Station get a unique view of planet Earth. In a single 90-minute orbit, they can see weather systems like hurricanes, man-made developments like cities and geological phenomena like volcanoes. Of all the planets NASA has explored, none yet have matched the dynamic complexity of our own Earth.  Luckily for us here on the ground, astronauts like Rick Mastracchio and Koichi Wakata have documented and shared their views from the orbiting laboratory 250 miles above the Earth. Both Mastracchio and Wakata tweeted during their 188 days in orbit, sharing photos of auroras, islands, lakes, bridges, glaciers, cities, deserts, mountains, clouds, and more. Photos like theirs give us a new perspective on our planet.  A fleet of NASA satellites also monitor the Earth’s vital signs every day. One example is the joint US-Japan Global Precipitation Measurement (GPM) mission, which launched while Mastracchio and Wakata were living on ISS. The astronauts briefed President Obama on GPM in April, describing it as “a groundbreaking satellite that is changing how we measure rain and snowfall from space.” Mastracchio and Wakata will be joined by Eric Holthaus, a meteorologist who writes about weather and climate for Slate’s Future Tense blog, and Gail Skofronick Jackson, a GPM scientist at NASA’s Goddard Space Flight Center.  You may submit questions on Twitter and Google+ in advance and during the Hangout using the hashtag #askNASA .Observing Earth from Space2014-09-17 20:30:001654 

Artur Popławski has been shared in 123 public circles

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AuthorFollowersDateUsers in CircleCommentsReshares+1Links
Richard Green93,025Engagers Showcase Circle, February 1, 2015If I sent you a notification, it means that you are included in my Engagers Showcase Circle. “Showcase” means that you are invited to leave a comment (on the original post) with a link to one of your own posts, which ideally should be one of your best recent posts.This circle consists of people who have engaged with one of my recent posts in the form of +1s, comments and reshares.Everyone mentioned below is also included in the circle.Ramanujan's nested radicalhttps://plus.google.com/101584889282878921052/posts/3WmvWEHyMNBThe exceptional symmetryhttps://plus.google.com/101584889282878921052/posts/ioQW2zGjwwMThe mystery of the missing areahttps://plus.google.com/101584889282878921052/posts/QU8aYaTCufqSunrisehttps://plus.google.com/101584889282878921052/posts/9QWbSALP2XUShakespearean Logichttps://plus.google.com/101584889282878921052/posts/MBqnMRgBiBJ“Nines” by Eric Standleyhttps://plus.google.com/101584889282878921052/posts/aR5BF9uV5n8Cherry pi (reshared from +David Richeson)https://plus.google.com/101584889282878921052/posts/bixJ7eGk3QmHappy New Year!https://plus.google.com/101584889282878921052/posts/ZJYFj1RogaSThe mathematics of card shufflinghttps://plus.google.com/101584889282878921052/posts/fAS8Y3YccfsThe sky and the fork in the path immediately preceding the arrival of the ice rinks of doomhttps://plus.google.com/101584889282878921052/posts/2MiqiTJ1FeuiPad landscape art (reshared from +Paul Haworth)https://plus.google.com/101584889282878921052/posts/27EwU49z1g8The fractional chromatic number of the planehttps://plus.google.com/101584889282878921052/posts/VbBk9JrLxqmDull (in Scotland) and Boring (in Oregon)https://plus.google.com/101584889282878921052/posts/ZuMzApfSPR4Partition and sum is fasthttps://plus.google.com/101584889282878921052/posts/Ad1ism1vJpJThe tautological clockhttps://plus.google.com/101584889282878921052/posts/diAyvxM7NuwMathematical Mr Menhttps://pl2015-02-01 15:55:59476180123217
Ichigo Kurosaki831Circle of awesomeness. Please followhttps://plus.google.com/u/0/b/116338847187157094371/115447032474133435721/postswhen you see the message ;)2014-12-09 02:16:58449616
Richard Green81,215Engagers Showcase Circle, September 14, 2014If I sent you a notification, it means that you are included in my Engagers Showcase Circle. “Showcase” means that you are invited to leave a comment (on the original post) with a link to one of your own posts, which ideally should be one of your best recent posts.This circle consists of people who have engaged with one of my recent posts in the form of +1s, comments and reshares. I skipped over one post because it received too much engagement, but I'm including a link to it for completeness.Everyone mentioned below is also included in the circle.Do nuclear physicists have half life crises?https://plus.google.com/101584889282878921052/posts/ayw6WPGGaFESt Peter's Church, Heyshamhttps://plus.google.com/101584889282878921052/posts/9DEtmbdz15zSmiling cow?https://plus.google.com/101584889282878921052/posts/9NuqPpsgtBkThe look-and-say sequence and Conway's Cosmological Theoremhttps://plus.google.com/101584889282878921052/posts/jEQ7zxFpJt4Cordyline australis, the “cabbage tree”https://plus.google.com/101584889282878921052/posts/hktDAgyo6mA“Maurits, stop picking at it. You'll only make it worse.” by David Swarthttps://plus.google.com/101584889282878921052/posts/gj327Ywh33T“Phyllotactic Portrait of Fibonacci” by Robert Boschhttps://plus.google.com/101584889282878921052/posts/8LykdvHpRvPFountain in Williamson Parkhttps://plus.google.com/101584889282878921052/posts/HzZTLQaQ9RT“Youth” by Silvia Cordeddahttps://plus.google.com/101584889282878921052/posts/T2Lo3c2zLxvThe arithmetic derivative, the Goldbach conjecture, and the twin prime conjecturehttps://plus.google.com/101584889282878921052/posts/9nY35Ma1pbUGlobe Thistlehttps://plus.google.com/101584889282878921052/posts/i8mtiyVikWhTallinn (reshared from +Paul Harper)https://plus.google.com/101584889282878921052/posts/U4DAQxK5fkxCubes passing in the night (reshared from +Sean Walker)https://plus.google.com/101584889282878921052/posts/32JKvAFqP9SThe graph of arctanhttps://plus.google.com/101584889282878921052/posts/FLvyDupud1z“Hopf Knott” by Peter Sittner2014-09-14 15:25:57463224129234
Becky Collins15,950Dance Related Circle :Circle of very #social #engagerspeople and companiesTo be included in my shares (#sharedcircle), be so kind to:1 - Do +1 t the post2 - Comment the post and specify your "category" (job or interest) Ex: Fashion, SEO, Companies, Social Media Marketing, Sailing, Photography, Bloggers/Writers, Web graphics and design, Italy, Artists, Sport, Finance/Economy ...3 - include the circle among your circles4 - share the circle (include yourself)Improve your popularity, be social be cool !Keep yourself updated, enjoy the Shared Circles Hellenic Alliance, you can share your shared circles inside the upcoming Community:https://plus.google.com/communities/112552559573595396104  #socialmedia  #media  #circles   #circleshare   #circlesharing  #circlecircle   #beckyscircle   #sharedcircles   #sharedpubliccircles  #sharedcircleoftheday  +Becky Collins ?2014-09-02 05:00:4347130935
Kukuh Malindo2322014-08-02 10:20:4450015218
Doug Hyatt5,196THIS IS MEGA CIRCLE 5    #DOUGHYATTCIRCLESHARING IT ADDED 1,000+ FOLLOWERS FOR ME IN A WEEK.1) ADD THE CIRCLE2) RESHARE THE CIRCLE AND BE SURE TO REMEMBER TO CHECK THE BOX WHICH INCLUDES YOURSELF.3)RESHARE TO PUBLIC EVERY DAY!BE SURE TO COPY THE HASHTAG LIST AND ADD IN YOUR COMMENTS LIKE I HAVE!  IT WILL GIVE YOU MAXIMUM RESULTS#doughyattcirclesharing#circles   #circlemania   #circletoday   #circleplus   #circleshare   #circlesharing   #CircleSharing   #sharecircle   #sharecirclethis   #worldsharecircle   #pleasecircleaddme   #circleoftheday   #newgenerationcircle  #monstercircle   #usacircle   #bestofthecircle   #shareofthecircle   #pleasecircle   #newbestcircl  add circle add circle * best share circle * share please pleaseplease add circle * best share circle * share please please#add #circle   #addcircle   #pleasecircle   #bestcircle   #mostcircle  #bestsharecircle   #newcircle   #top100circle   #best500circle   #top500circle   2014-07-16 15:49:10477151421
Richard Green74,896Engagers Showcase Circle, July 12, 2014If I sent you a notification, it means that you are included in my Engagers Showcase Circle. “Showcase” means that you are invited to leave a comment (on the original post) with a link to one of your own posts, which ideally should be one of your best recent posts.This circle consists of people who have engaged with one of my recent posts in the form of +1s, comments and reshares.Everyone mentioned below is also included in the circle.Thistlehttps://plus.google.com/101584889282878921052/posts/CKB7ks1VWZ3“Star Gate” by +Gary Matthews https://plus.google.com/101584889282878921052/posts/ean83YFix7sBubble (reshared from +Ann-Marie Jurek)https://plus.google.com/101584889282878921052/posts/PkybKJfXv14Gabriel's Hornhttps://plus.google.com/101584889282878921052/posts/MXNTuq8ZEBPHibiscus flowerhttps://plus.google.com/101584889282878921052/posts/Xsmw62v98CHStackable 12-sided dicehttps://plus.google.com/101584889282878921052/posts/5mLoukdZAA5Astronomical book priceshttps://plus.google.com/101584889282878921052/posts/S15LTNxJA9GPyritohedral symmetryhttps://plus.google.com/101584889282878921052/posts/XQUYMrKGiAQSt Stephen's Basilica (reshared from +Laura Orange)https://plus.google.com/101584889282878921052/posts/ApnSMBsUWFeCircular Pascal Arrays and Fibonacci numbershttps://plus.google.com/101584889282878921052/posts/NVqYKoRVnHg“Fungi tree” by +Tom Beddardhttps://plus.google.com/101584889282878921052/posts/J6S4bDc6QwQAs always, reshares of this circle are appreciated, and I look forward to seeing everyone's links. Thanks for reading my posts!2014-07-12 05:31:36476209123224
Greg Strandberg942Sci-Fi and Fantasy Readers CircleWant to connect with #fantasyauthor  and #scifiauthor  ?This #circleshare  has those folks and chances are you'll get some interesting things in your feed. #circlesharing   #googleshare   #circleoftheday   #sharedcircles   #fantasywriters   #sciencefictionfantasy   #scifi  2014-07-10 23:13:52173000
Doug Hyatt3,234MANY NEW PEOPLE #2 THAT WILL PLUS 1 and RESHARESimply ADD yourself to this circle, copy the #HASHTAGS, and REPOST#circles   #circlemania   #circletoday   #circleplus   #circleshare   #circlesharing   #CircleSharing   #sharecircle   #sharecirclethis   #worldsharecircle   #pleasecircleaddme   #circleoftheday   #newgenerationcircle  #monstercircle   #usacircle   #bestofthecircle   #shareofthecircle   #pleasecircle   #newbestcircl  add circle add circle * best share circle * share please pleaseplease add circle * best share circle * share please please#add #circle   #addcircle   #pleasecircle   #bestcircle   #mostcircle  #bestsharecircle   #newcircle   #top100circle   #best500circle   #top500circle   #circleoftheday   #circledaily   #nicecircle   2014-06-29 10:59:0549121717
Dina Tika0Here is a group of Active Engagers, Circle Sharers, Awesome Plus Oners, and Cool People on Google Plus!   Circle Sharing is an awesome way to increase your followers and active engagers on your profile. Some of my favorite people that I've met here on Google + through Circle Sharing.    Want to be in the next Circle of Awesomeness? Follow the Steps Below!  ☛ Add the circle ☛ Share in the Public ☛ Plus 1 the Post. ☛ Comment. 2014-06-10 05:53:52479001
Lynda Chervil534This circle is for +++++++Green Living folks++++++. You'll find green minded individuals who are committed to promoting an environmentally conscious approach to living, technology, business etc. Please feel free to add all of these fine folks and share this circle with your friends! Just add the circle and share it with your followers! #greenliving   #environmentalism   #environment   #alternativeenergy  2014-05-30 19:27:5849917717
Richard Green68,138Engagers Showcase Circle, May 12, 2014If you received a notification, it means that you are included in my Engagers Showcase Circle. “Showcase” means that you are invited to leave a comment (on the original post) with a link to one of your own posts, which ideally should be one of your best recent posts.This circle consists of people who have engaged with one of my recent posts in the form of +1s, comments and reshares.Everyone mentioned below is also included in the circle.Peruvian glacier (reshared from +Irina Tcherednichenko)https://plus.google.com/101584889282878921052/posts/Z6NR7zaZa5qGrape hyacinthshttps://plus.google.com/101584889282878921052/posts/GZoHH2bnRUiSymmetries of a regular octagonhttps://plus.google.com/101584889282878921052/posts/bFcGcsurF5HHappy Easter!https://plus.google.com/101584889282878921052/posts/YkWFA9WmxVAPerfect Squared Squareshttps://plus.google.com/101584889282878921052/posts/HDU4mWD3wptOur dog Rexhttps://plus.google.com/101584889282878921052/posts/5JtEcnt5MpPPerrott's Follyhttps://plus.google.com/101584889282878921052/posts/8DzXHLn6fe4The Lonely Runner Conjecturehttps://plus.google.com/101584889282878921052/posts/YHJkEdKg3jz“Owl” by Andrea Mininihttps://plus.google.com/101584889282878921052/posts/12uh9J9MjMXPenrose nanotileshttps://plus.google.com/101584889282878921052/posts/7PZPSLaFkAx“Interlaced Stars” by Magnenauthttps://plus.google.com/101584889282878921052/posts/5vNjv1yjVMqBlack Forest View (reshared from +Ralph Reichert)https://plus.google.com/101584889282878921052/posts/YKPsvCFDyUv“Hippie Bus Apartments” by +Richie Montgomeryhttps://plus.google.com/101584889282878921052/posts/iCMXCBVzjd2Not crazy enoughhttps://plus.google.com/101584889282878921052/posts/d5EeDSoHe5u“Impossible Sphere” by Alexey Ermushevhttps://plus.google.com/101584889282878921052/posts/J2yUMw6YASqBeethoven, high on 2014-05-12 05:07:06489214128231
Bakerson, LLC287This Circle is Amazing! If you want more likes, shares, and +1's, you will want to join and add this circle! To be added to this circle of Super Engagers simply +1, reshare, and comment!! You will be added to the next go around soon. Boost your social status quickly!2014-05-01 15:52:48479121014
Krzysztof Skomra5,314*PEOPLE WHO SAY "THANK YOU"  This circle is to show appreciation to the who said thanks to my last circle#sharedcircle #publiccircleshare   #circleshare   #circleoftheday #circleshares   #circlesharing   #circlesshare   #publicsharedcircle #publicsharedcircles   #sharedcircleoftheday #sharedcircleoftheweek #sharedcircles   #sharedpubliccircle #todayspublicsharedcircle #todayssharedcircle   #publiccircle #circle   #circles   #awesome #awesomeness #awesomepeople #shareyourcircle   #bestengagers #followers   #followback #paulaawesomecircle   #awesomecircles #paulateshima #topsharedcircle #topsharedcircle   #myseoissocial #besocial   #socializethesocial   #trust #circles #shar2014-04-19 06:31:05377139
Krzysztof Skomra5,314*PEOPLE WHO SAY "THANK YOU"  This circle is to show appreciation to the who said thanks to my last circle#sharedcircle #publiccircleshare   #circleshare   #circleoftheday #circleshares   #circlesharing   #circlesshare   #publicsharedcircle #publicsharedcircles   #sharedcircleoftheday #sharedcircleoftheweek #sharedcircles   #sharedpubliccircle #todayspublicsharedcircle #todayssharedcircle   #publiccircle #circle   #circles   #awesome #awesomeness #awesomepeople #shareyourcircle   #bestengagers #followers   #followback #paulaawesomecircle   #awesomecircles #paulateshima #topsharedcircle #topsharedcircle   #myseoissocial #besocial   #socializethesocial   #trust #circles #shar2014-04-19 06:30:35377116
Krzysztof Skomra5,314*PEOPLE WHO SAY "THANK YOU"  This circle is to show appreciation to the who said thanks to my last circle#sharedcircle #publiccircleshare   #circleshare   #circleoftheday #circleshares   #circlesharing   #circlesshare   #publicsharedcircle #publicsharedcircles   #sharedcircleoftheday #sharedcircleoftheweek #sharedcircles   #sharedpubliccircle #todayspublicsharedcircle #todayssharedcircle   #publiccircle #circle   #circles   #awesome #awesomeness #awesomepeople #shareyourcircle   #bestengagers #followers   #followback #paulaawesomecircle   #awesomecircles #paulateshima #topsharedcircle #topsharedcircle   #myseoissocial #besocial   #socializethesocial   #trust #circles #shar2014-04-19 06:29:49377335
Krzysztof Skomra5,314*PEOPLE WHO SAY "THANK YOU"  This circle is to show appreciation to the who said thanks to my last circle#sharedcircle #publiccircleshare   #circleshare   #circleoftheday #circleshares   #circlesharing   #circlesshare   #publicsharedcircle #publicsharedcircles   #sharedcircleoftheday #sharedcircleoftheweek #sharedcircles   #sharedpubliccircle #todayspublicsharedcircle #todayssharedcircle   #publiccircle #circle   #circles   #awesome #awesomeness #awesomepeople #shareyourcircle   #bestengagers #followers   #followback #paulaawesomecircle   #awesomecircles #paulateshima #topsharedcircle #topsharedcircle   #myseoissocial #besocial   #socializethesocial   #trust #circles #shar2014-04-19 06:29:03377157
Krzysztof Skomra5,201PEOPLE WHO SAY "THANK YOU"  This circle is to show appreciation to the who said thanks to my last circle#sharedcircle #publiccircleshare   #circleshare   #circleoftheday #circleshares   #circlesharing   #circlesshare   #publicsharedcircle #publicsharedcircles   #sharedcircleoftheday #sharedcircleoftheweek #sharedcircles   #sharedpubliccircle #todayspublicsharedcircle #todayssharedcircle   #publiccircle #circle   #circles   #awesome #awesomeness #awesomepeople #shareyourcircle   #bestengagers #followers   #followback #paulaawesomecircle   #awesomecircles #paulateshima #topsharedcircle #topsharedcircle   #myseoissocial #besocial   #socializethesocial   #trust #circles #sharedcircle 2014-04-15 11:37:45501006
Krzysztof Skomra5,201PEOPLE WHO SAY "THANK YOU"  This circle is to show appreciation to the who said thanks to my last circle#sharedcircle #publiccircleshare   #circleshare   #circleoftheday #circleshares   #circlesharing   #circlesshare   #publicsharedcircle #publicsharedcircles   #sharedcircleoftheday #sharedcircleoftheweek #sharedcircles   #sharedpubliccircle #todayspublicsharedcircle #todayssharedcircle   #publiccircle #circle   #circles   #awesome #awesomeness #awesomepeople #shareyourcircle   #bestengagers #followers   #followback #paulaawesomecircle   #awesomecircles #paulateshima #topsharedcircle #topsharedcircle   #myseoissocial #besocial   #socializethesocial   #trust #circles #sharedcircle 2014-04-15 11:36:23501012
Krzysztof Skomra5,201PEOPLE WHO SAY "THANK YOU"  This circle is to show appreciation to the who said thanks to my last circle#sharedcircle #publiccircleshare   #circleshare   #circleoftheday #circleshares   #circlesharing   #circlesshare   #publicsharedcircle #publicsharedcircles   #sharedcircleoftheday #sharedcircleoftheweek #sharedcircles   #sharedpubliccircle #todayspublicsharedcircle #todayssharedcircle   #publiccircle #circle   #circles   #awesome #awesomeness #awesomepeople #shareyourcircle   #bestengagers #followers   #followback #paulaawesomecircle   #awesomecircles #paulateshima #topsharedcircle #topsharedcircle   #myseoissocial #besocial   #socializethesocial   #trust #circles #sharedcircl2014-04-15 11:35:35501012
Julek Maciejewski0PEOPLE WHO SAY "THANK YOU"  This circle is to show appreciation to the 411  who said thanks to my last circle, and 88 new #thankyou members entering this circle for the first time. All of these people I observed over the last week taking a second to say "thanks". I was so pleased, I created a circle of 500 of you. I am amazed every day what a fun group of people have been accumulating here on Google+. I continue to be impressed with quality, the professionalism,  and the sincerity of everyone I am meeting. This circle is for YOU  (Click on this #thankyoumagic to see my other 2014 circles) #Circlemagic #Magiccircle   PLEASE VISIT the "The Magic of Circles" hosted by +Circle Magic Each day I am gathering the most active Circles I can find. Circles are #Magic , and if you love them, they will love you back.   PLEASE SHARE A #CIRCLE  Do you want to Grow your ? Get started now, add or share a Circle and watch your  grow.  Sharing circles benefits everyone, Sharing circles helps you by getting you added to their next circle and it helps the "Circle Master" by getting them recognized and promoted as active circle distributors. The more shares he/she gets THIS WEEK the more people that follow you NEXT WEEK.THE MAGIC OF CIRCLEShttps://plus.google.com/u/1/b/103678492966360900254/communities/110533309056060602888SPECIAL THANK YOU FROM +Circle Magic  - I just wanted to take a second to thank all the best of the best circle promoters, who probably mostly go un-noticed, but without them, the success of the circle sharing would be much more challenging. These SPECIAL people have all dropped by my community "The Magic of Circles" and shared someone else's circle. #TIPHAT to .....+Logan Lynn Roberts+Michael Q Todd+Sandra Deichmann...SPECIAL THANK YOU TO THE CIRCLE MASTERS - I cannot forget you either. without your  thankless hard work, building, maintaining, auditing, promoting these lists in the first place, none of us would have anything to promote.+Selected Circles 2014...Here are some of the most amazing circles I have seen released in the last seven days, each of these amazing people have worked really hard to deliver a specialize list just for you+Chad Hart...FAVOR TO ASK - As if all of you are not already doing TOO much work already. I was  hoping to ask, that when you drop of your circles each day, that you take an extra  minute to SHARE a a couple other peoples circles too. My team tries to give each and every circle between 7-12 shares to every circle dropped off in my community "The Magic of Circles" but as we become more and more successful we find we are getting  20-30 new circles listed each day, and it is getting harder and harder to keep up with everyone, and already I am starting to have to pick and choose which circles I  can afford to run #Empireavenue   missions on to get that extra 5-6 shares on the most active  circles..I know we are all doing this for free, with no real idea of why we are doing this,  except that it feels li2014-04-14 01:08:56501017
Julek Maciejewski0PEOPLE WHO SAY "THANK YOU"  This circle is to show appreciation to the 411  who said thanks to my last circle, and 88 new #thankyou members entering this circle for the first time. All of these people I observed over the last week taking a second to say "thanks". I was so pleased, I created a circle of 500 of you. I am amazed every day what a fun group of people have been accumulating here on Google+. I continue to be impressed with quality, the professionalism,  and the sincerity of everyone I am meeting. This circle is for YOU  (Click on this #thankyoumagic to see my other 2014 circles) #Circlemagic #Magiccircle   2014-04-14 01:05:48501112
Circle Connection570The Mix it Up Circle! #circleconnection  2014-04-10 20:22:45481723035
Richard Green62,178Engagers Showcase Circle, March 29, 2014If you received a notification, it means that you are included in my Engagers Showcase Circle. “Showcase” means that you are invited to leave a comment (on the original post) with a link to one of your own posts, which ideally should be one of your best recent posts.This circle consists of people who have engaged with one of my recent posts in the form of +1s, comments and reshares. My most recent post will be included next time.Everyone mentioned below is also included in the circle.Happy Pi Day!https://plus.google.com/101584889282878921052/posts/74oomcTuJoVSierpinski sponge (reshared from +David Roberts)https://plus.google.com/101584889282878921052/posts/ZKUreVJfWxoCloud Gate (reshared from +Christian Cable)https://plus.google.com/101584889282878921052/posts/XGRNqbbBjgEContinued fractionshttps://plus.google.com/101584889282878921052/posts/2mbZvUrewmCFood mapshttps://plus.google.com/101584889282878921052/posts/BKMgJ6WzfiyNeptune's Lament (reshared from +Paul Haworth)https://plus.google.com/101584889282878921052/posts/1AHZMatxm8JEscher's “Ascending and Descending”... in Legohttps://plus.google.com/101584889282878921052/posts/9XJsanAguNCDead Horse Point State Park (reshared from +Tom Malloy)https://plus.google.com/101584889282878921052/posts/edKVZ8NqAW4It's the Chicken from Hell! (reshared from +Chad Haney)https://plus.google.com/101584889282878921052/posts/2tc9WchRABJDerangements, or how everyone got the wrong homework backhttps://plus.google.com/101584889282878921052/posts/Bjb3CvDYBwxSnowy La Sal Mountains (reshared from +Tom Malloy)https://plus.google.com/101584889282878921052/posts/8tiW4gbzVrXMultimove chess2014-03-29 04:30:56482206156228
Bob Mulholland20,590Haven't done a  #FollowFriday  in a while. Beam these mofos into your Circles , Scotty!Nothin' but a buncha  #StarTrek  nerds in here.... #SharedCircles  2014-03-07 20:06:14486519
Artur M.35,611circle of people, with active life position in Google+2014-01-19 04:03:28438251674
Richard Green52,056Engagers Showcase Circle, January 9, 2014If you received a notification, it means that you are included in my Engagers Showcase Circle. “Showcase” means that you are invited to leave a comment (on the original post) with a link to one of your own posts, which ideally should be one of your best recent posts.This circle consists of people who have engaged with one of my recent posts in the form of +1s, comments and reshares. I waited too long to share the circle, so I ran out of room. As a consequence, several of my recent posts will be included next time instead.Everyone mentioned below is also included in the circle.50,000 followershttps://plus.google.com/101584889282878921052/posts/CXAHuH8epQaHyperbolic honeycombs (reshared from +Roice Nelson)https://plus.google.com/101584889282878921052/posts/XkTecGK6CghMarie Antoinette: Let Them Eat Niphttps://plus.google.com/101584889282878921052/posts/hu9YeJxPVPmMy daughter jugglinghttps://plus.google.com/101584889282878921052/posts/F38QyMXs7YDBridge photography (reshared from +Eric Larson)https://plus.google.com/101584889282878921052/posts/XQsEw9CtLjSWhite-throated Kingfisher (reshared from +Anuradha Goyal)https://plus.google.com/101584889282878921052/posts/H6zMhGj3aS5What's up with 163?https://plus.google.com/101584889282878921052/posts/7enyPxZW3RBSeascape (reshared from +Paul Haworth)https://plus.google.com/101584889282878921052/posts/3XxHeKz61wcLittlewood's Law of Miracleshttps://plus.google.com/101584889282878921052/posts/gLmKu3vPXhsKnot theorists of the world untie!https://plus.google.com/101584889282878921052/posts/HLmx56CSPDEMelted crayon artwork (reshared from +Laura Ockel)https://plus.google.com/101584889282878921052/posts/FkeBeRvtnRR5-dimensional hypercubehttps://plus.google.com/101584889282878921052/posts/9soBKs4TtRmAs alwa2014-01-09 22:16:33499193142241
Richard Green50,191Engagers Showcase Circle, December 27 2013If you received a notification, it means that you are included in my Engagers Showcase Circle. “Showcase” means that you are invited to leave a comment (on the original post) with a link to one of your own posts, which ideally should be one of your best recent posts.This circle consists of people who have engaged with one of my recent posts in the form of +1s, comments and reshares. I skipped two of the posts below because they attracted low-quality engagement, but I'm listing them here for completeness.Everyone mentioned below is also included in the circle.Seven cylinders a-touchinghttps://plus.google.com/101584889282878921052/posts/FGZHF7kqaWkLeaves in Water (reshared from +Marjolein Caniels)https://plus.google.com/101584889282878921052/posts/bQvdz7yL82FElf on the Shelf (on a Reindeer)https://plus.google.com/101584889282878921052/posts/DvXFqZ89andIridescent Cloudshttps://plus.google.com/101584889282878921052/posts/S2izLcENvjnSpear Lodge Manhttps://plus.google.com/101584889282878921052/posts/DUWKF4CG3XHLaser Christmas decorationshttps://plus.google.com/101584889282878921052/posts/B7ngNnfdQ6bRadial Normals (by +Patrick Honner) https://plus.google.com/101584889282878921052/posts/34zwPWe3cD2Rainbow photography (reshared from +James Wheeler)https://plus.google.com/101584889282878921052/posts/Ph8JSPXWAEnCornwall seascape painting (reshared from +Paul Haworth) https://plus.google.com/101584889282878921052/posts/BypBhhkPzMcMulticultural Santa (reshared from +Mike Stay)https://plus.google.com/101584889282878921052/posts/Uk9AreT7eC5Temari ballshttps://plus.google.com/101584889282878921052/posts/E5UibQyB1rjThe Eagle Catcherhttps://plus.google.com/101584889282878921052/posts/BXayNs16RdKChristmas lights in Boulder2013-12-27 07:02:36454179145235
Chill Ideaz Jay Beek Rocksuperstar0Another round of #topengagers  !(Be sure to click READ MORE, it is important if you want to be included in this circle next time)These people shareThese people +1These people create original contentThese people make G+ BetterIf you add this circle, you will, without a doubt, see your engagement soar through the roof.  Guaranteed.  If you were notified, you guessed it, you are already in.To be included in the NEXT top engagers circle this week is easier than ever.1.) Follow me2.) Share this post3.) Keep engaging#sharedcircle   #sharedcircles   #sharecircle   #publiccircle   #publiccircles   #publicsharedcircles   #topengagers   #engagers   #snowballcircle   #snowball   #circles   #circleshare   #circlemasters   #circlemaster   #circlequeen   #crazycircles   #circlehub   #morefollowers #ghosttown 2013-11-25 00:30:29405006
Chill Ideaz Jay Beek Rocksuperstar0Another round of #topengagers  !(Be sure to click READ MORE, it is important if you want to be included in this circle next time)These people shareThese people +1These people create original contentThese people make G+ BetterIf you add this circle, you will, without a doubt, see your engagement soar through the roof.  Guaranteed.  If you were notified, you guessed it, you are already in.To be included in the NEXT top engagers circle this week is easier than ever.1.) Follow me2.) Share this post3.) Keep engaging#sharedcircle   #sharedcircles   #sharecircle   #publiccircle   #publiccircles   #publicsharedcircles   #topengagers   #engagers   #snowballcircle   #snowball   #circles   #circleshare   #circlemasters   #circlemaster   #circlequeen   #crazycircles   #circlehub   #morefollowers #ghosttown 2013-11-24 16:58:13405007
Chill Ideaz Jay Beek Rocksuperstar0Another round of #topengagers  !(Be sure to click READ MORE, it is important if you want to be included in this circle next time)These people shareThese people +1These people create original contentThese people make G+ BetterIf you add this circle, you will, without a doubt, see your engagement soar through the roof.  Guaranteed.  If you were notified, you guessed it, you are already in.To be included in the NEXT top engagers circle this week is easier than ever.1.) Follow me2.) Share this post3.) Keep engaging#sharedcircle   #sharedcircles   #sharecircle   #publiccircle   #publiccircles   #publicsharedcircles   #topengagers   #engagers   #snowballcircle   #snowball   #circles   #circleshare   #circlemasters   #circlemaster   #circlequeen   #crazycircles   #circlehub   #morefollowers #ghosttown 2013-11-24 08:00:39405105
Chill Ideaz Jay Beek Rocksuperstar0#sharedcircle   #sharedcircles   #sharecircle   #publiccircle   #publiccircles   #publicsharedcircles   #topengagers   #engagers   #snowballcircle   #snowball   #circles   #circleshare   #circlemasters   #circlemaster   #circlequeen   #crazycircles   #circlehub   #morefollowers #ghosttown2013-11-24 06:33:32405005
Chill Ideaz Jay Beek Rocksuperstar0This group will follow back !#sharedcircle   #sharedcircles   #sharecircle   #publiccircle   #publiccircles   #publicsharedcircles   #topengagers   #engagers   #snowballcircle   #snowball   #circles   #circleshare   #circlemasters   #circlemaster   #circlequeen   #crazycircles   #circlehub   #morefollowers #ghosttown2013-11-24 01:22:28405104
Chill Ideaz Jay Beek Rocksuperstar0Here's the latest update of my Engagers circle. The last version of this circle was close to full, so I started it again from scratch. This circle includes people who have recently engaged with one of my posts in the form of +1s, comments and public reshares.I might have missed some people because of limitations of the Google+ interface. Fortunately, the next release of +Circloscope on June 28 is likely to include new features that make circles like this one much easier to create.I've included 100 guest members in this circle, taken from my Favourites circle, which I haven't shared in a while.Like my previous circle share, this is a Showcase circle! What this means is that everyone is invited to leave a comment on (the original post of) this circle share containing a link to one of your own posts. Ideally, this should be something that you posted recently and that you are particularly pleased with. (Don't post spam though; I will delete it.)2013-11-24 00:22:24405014
Silvio De Rossi3,801It's dance time! #saturdaynight  Circle! :DThis is my first #sharedcircle of #topengagers, a group of interesting people who share unique and original content.What are you waiting to add them to your circles? :)To be included in my shares (#sharedcircle), be so kind to:1) Do +1 t the post2) Comment the post and specify your "category" (ex: fashion, photography, seo, social media marketing)3) Include the circle among your circles4) Share the circle (include yourself)5) Smile to life, sun in sunshine.Special thanks my dear #friends and #followers:+Raffaele Romeo Arena  +Tancredi Matranga +Alessandro Folghera +Christine DeGraff +Viktor Mulin +Cesare Riccardo Have a good life!#topsharedcircle   #facciamolastoria   #leonardo   #deroplus   #circleoftheday   #sharedcircle   #topsharedcircle   #sharedcircleoftheday   #myseoissocial   #besocial   #socializethesocial  #trust #circles  #circleshare  #sharedcircle   2013-10-26 19:28:02457531239
Raffaele Romeo Arena5,311Welcome my dear #friends and #follower this is #thursdaycircle #topengagers  This circle is very important because it's composed of many important peolple, that often #interact with me and our community.You remember that #social  is   #sharedlife  and #follow is #friendship  . And so:-include me in your circles-create a large network-shared the circle (include yourself)-add +1 to the post-interact will allow you to enter the great circle that soon will share-happy life and sun is sunshine.Special thanks, to my #friends  and #followers  :+Alessandro Folghera +Mario Falcetti +Cesare Riccardo +Edith Kukla +Dominique Keller +Viktor Mulin +Silvio De Rossi +Tancredi Matranga +Tazein Mirza Saad +Fashion Circles #circles   #circleshare   #sharedcircle 2013-10-24 09:20:30459623468
Circle Construct is for communication561A little room to add yourself. Please public share, because if you are not in this circle, that is how you will get included into future circles somewhere. 2013-10-22 08:23:1946613833
Refurio Anachro1,560Mochizuki club circle Here's everyone who showed interest (posted) when Shinichi Mochizuki announced he had proof for the abc- conjecture. It's been great fun to read the impressions from back then.I have seen especially interesting posts by +John Baez, who gave very accessible first fragmentary impressions. Just search for his name and "mochizuki" and don't forget to read the comments!Also, i urge you to look at +lieven lebruyn's profile, who may be the one who got deepest into explaining it. And look: Colored, hand drawn diagrams! He's been posting MinuteMochizuki on his blog until mid of june and i think he'll be back!http://matrix.cmi.ua.ac.be/ For more here's the polymath page which, among many other interesting links, lists quite a reaction here on google+. +Jordan Ellenberg +Timothy Gowers +Terence Tao  also made it into this circle.http://michaelnielsen.org/polymath1/index.php?title=ABC_conjectureAs for where matters stand, it may be that Mochizuki is currently giving private lectures to Go Yamashita, a colleague, who might be inclined to do the travelling to explain it to others. Or so... that's what this blog post by Caroline Chen suggests:http://projectwordsworth.com/the-paradox-of-the-proof/Minhyong Kim, a friend of Mochizuki at Oxford, has been posting on mathoverflow an early attempt to give some overview from a closer vantage point:http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjectureAnd, as i must say, i stole the following taglist from +Rusty Ferguson's latest circle share. You wrote some appealing stuff, your tags must be good:https://plus.google.com/112078832938393562132/posts/WvqYjBrKiir #sharedcircles   #publicsharedcircles   #circleshare   #sharedpublicricle   #circlesharing   #circle   2013-10-18 00:17:00125145
Richard Green32,670Engagers Showcase Circle, October 16 2013If you received a notification, it means that you are included in my Engagers Showcase Circle. “Showcase” means that you are invited to leave a comment (on the original post) with a link to one of your own posts, which ideally should be one of your best recent posts.This circle consists of people who have engaged with one of my recent posts in the form of +1s, comments and reshares. In the last two weeks, I discussed mathematical aspects of the art of M.C. Escher, as well as the fractal known as the Sierpinski triangle. I also posted about a sign with a typo on it, and a set of tea mugs for the kind of people who would be annoyed by such a mistake. I posted some original photography; one of my photo posts got into What's Hot, so I skipped over it when compiling this circle. I also reshared posts by +John Baez, +Arnold Hayes, +Jason Hill, and +Bo Insogna, all of whom are included in the circle.Google's persistent domain name bug has meant that I wasn't able to include everyone I wanted to in this circle. The bug is now even worse than before, and has stripped certain people out of all of my circles because of my attempt to add them to this one! I hope they fix this soon, because it's really annoying.As always, reshares of this circle are appreciated, and I look forward to seeing everyone's links. Thanks for reading my posts!2013-10-16 17:57:37418246152238
Raffaele Romeo Arena4,418#mondaysharedcircle in your #circle #seocircle #fashioncircle This #circle includes important followers to affirm a strong #community getting bigger. Join the community and shareTo be included in my shares (#sharedcircle), be so kind to:1 - Do +1 t the post2 - Comment the post and specify your "category" (job or interest)  Ex: Fashion, SEO, Companies, Social Media Marketing, Sailing, Photography, Bloggers/Writers, Web graphics and design, Italy, Artists, Sport, Finance/Economy ...3 - Include the circle among your circles4 - Share the circle (include yourself)If you come accross Google error messages, incorporating my circles, please provide me the error, I'm classifying these errors. Have a wonderful Monday and all week; and a better popularityMost important: Keep yourself updated, enjoy the Shared Circles Hellenic Alliance, you can share your shared circles inside the upcoming Community:https://plus.google.com/communities/112552559573595396104Special thaks to my friend and special follower:+Alessandro Folghera+Viktor Mulin+Fashion Circles+Scott Buehler+Sam Stormborn Ormandy+Владимир Самсонов+Ludovic Moreeuw+Dominique Keller+Tancredi Matranga+Silvio De Rossi+Edith Kukla2013-10-14 20:44:30460211238
Raffaele Romeo Arena3,803Astonishing #topengagers   Happy  #Tuesday my #friends , and you are welcome in my #tuesdaycircle A nice group of people who I found to share and comment my posts, and have as a goal to grow our #community.And so, after difficult day on #Monday we are to enjoy on Tuesday in #fashionstyle To be included in my shares (#sharedcircle), be so kind to:1 - Do +1 t the post2 - Comment the post and specify your "category" (job or interest)      (ex: fashion, photography, seo, social media marketing)3 - include the circle among your circles4 - share the circle (include yourself)5 - Plus and share the new important #fashionprofile https://plus.google.com/u/0/b/101905178080519621345/101905178080519621345/posts5 - smile to life, sun in sunshine.Thanks to all of you who are part of my circles and those who will be part of, in particular, I thank my #friends and #followers :+Alessandro Folghera+Alfredo Torres+Dirk Talamasca+Viktor Mulin+Cesare Riccardo+Mario Falcetti+Владимир Самсонов+Dominique KellerHave a wonderful weekend and increasing your #popularity and your #circles#topsharedcircle   2013-10-08 08:07:37463571660
Circle Construct is for communication561To You, When you are not comfortable with circles here is exactly what to do....1. Name the Circle and save the #circle .2. Give the circle creator credit #circleconstruct   3. Give credit to who shared the circle with you #circleconstruct   4. Add yourself to the circle. 5. Share the circle public, all your circles, or specific people who would      benefit from the people in the circle. 6. Say thank you in the stream. (comments)7. Let me know if you don’t wish to be in shared circles, we have          circles marked for that here so you can still receive page member  information.#circleconstruct  says you are welcome in advance.... #queenpld   #circleconstruct   #vipcirclemember    #highnumbers   #samepeople   #everycircle  2013-09-29 10:08:244681835
Richard Green27,236Engagers Showcase Circle, September 21 2013If you received a notification, it means that you are included in my Engagers Showcase Circle. “Showcase” means that you are invited to leave a comment (on the original post) with a link to one of your own posts, which ideally should be one of your best recent posts.This circle consists of people who have engaged with one of my recent posts in the form of +1s, comments and reshares. This week, I posted about the union-closed sets conjecture in mathematics, and I shared some of my original photography and some artwork from my daughter. I also reshared posts by +Paul Haworth, +Tiffany Lykins, +Jeni Ong, +Pauline Taylor and +Rupert Wood. I ran out of room on this circle because I waited too long to share it, so I excluded some people who didn't have me in their circles. However, everyone who reshared the last version of the circle should be included, provided that they have a profile picture. I did not include the engagers on my two most recent posts, but I will include them next time. I apologise to +Azure Tackert and +Eng UNG, who are supposed to be in this circle, but Google will not let me circle them back because of the annoying domain restriction bug.As always, reshares of this circle are appreciated, and I look forward to seeing everyone's links. Thanks for reading my posts!2013-09-21 16:59:27473238160248
Raffaele Romeo Arena2,188Welcome in my #weekendcircle .You remember that #social  is  #sharedlife  and #follow is #friendship  . And so:-include me in your circles-create a large network-shared the circle (include yourself)-add +1 to the post-happy lifeSpecial thanks, to my #friends  and #followers  :+Alessandro Folghera +Mario Falcetti +Cesare Riccardo  #circles   #circleshare  #sharedcircle #circlesharing   #follower #socialnetworking   #social   #followme   #follow4followback  2013-09-13 17:59:23468472555
Richard Green26,219Engagers Showcase Circle, September 13 2013If you received a notification, it means that you are included in my Engagers Showcase Circle. “Showcase” means that you are invited to leave a comment (on the original post) with a link to one of your own posts, which ideally should be one of your best recent posts.This circle consists of people who have engaged with one of my recent posts in the form of +1s, comments and reshares.This week, I have been mostly posting about: Ramsey theory in mathematics; oddly comprehensible sentences; the wave/particle nature of light; and the mathematics of electrical networks. I've also posted some original photography and reshared posts from +Marjolein Caniels, +AyJay Schibig and +Debashish Samaddar.As always, reshares of this circle are appreciated, and I look forward to seeing everyone's links. Thanks for reading my posts!2013-09-13 15:31:32467212117223
Raffaele Romeo Arena991#happycircleWelcome   in my #sharedcircles  in #wednesday and happy #sharedweek  Special thanks to +Yaiza Pozo Vara +Mario Falcetti +Alessandro Folghera You remember that #social  is #sharedlife  and #follow  is #friendship And so:-Include me in your circles-create a large network-share the circle (include yourself)-add +1 to the post-happy life#Wednesdaysharedcircle   #topsharedcircle   #circleoftheday   #sharedcircle #trustinme  #circlesharing   #circleshare        #circles        #circleoftheday   #sharedpubliccircles     #sharedcircles    #share  #vipsnowballcircle #sharedcircleoftheday        #sharewithyou        #circlefriday   #circlethursday  #followme     #followers #followback #circle #g2013-08-21 13:32:27446632681
Kai Saffron14,954Mathematician +Richard Green's shared circle of highly engaged Plussers!  Consider adding to liven up your stream-  Remember: "nothing ventured, nothing gained". #SharedCircles   #GooglePlus   #GhostTown   2013-08-07 16:37:49446582266
Leo Walsh3,691Richard Green originally shared:Engagers Showcase Circle, August 5 2013 If you received a notification, it means that you are included in my Engagers Showcase Circle. “Showcase” means that you are invited to leave a comment (on the original post) with a link to one of your own posts, which ideally should be one of your best recent posts.This circle consists of people who have engaged with one of my recent posts in the form of +1s, comments and reshares. I may have missed some people due to limitations of the G+ interface, but as usual, everyone who reshared last week's version of the circle has been included this time.As always, reshares are appreciated. And, if you have been, thanks for reading my posts!2013-08-06 00:51:27447521360
Richard Green20,938Engagers Showcase Circle, August 5 2013 If you received a notification, it means that you are included in my Engagers Showcase Circle. “Showcase” means that you are invited to leave a comment (on the original post) with a link to one of your own posts, which ideally should be one of your best recent posts.This circle consists of people who have engaged with one of my recent posts in the form of +1s, comments and reshares. I may have missed some people due to limitations of the G+ interface, but as usual, everyone who reshared last week's version of the circle has been included this time.As always, reshares are appreciated. And, if you have been, thanks for reading my posts!2013-08-05 23:36:22446267123229
Michael Q Todd37,249A circle of active engagersHave a great weekend hanging out with these people on Google Plus! #circleshare #circlesharing #circles  2013-08-02 21:35:0746914268170

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Most comments: 17

2015-01-09 22:08:17 (17 comments, 0 reshares, 19 +1s)Open 

No, bez jaj. Jeżeli ktoś wyraża zdanie, że wyznawcy islamu to potencjalni terroryści nienawidzący wartości jakim hołduje (bądź twierdzi, że hołduje) zachodnia cywilizacja, to nie oznacza, że jest rasistą. Można mu przypisać wiele - od w miarę łagodnych cech: ignorancji czy skłonności do nadmiernych uogólnień, po nienawiść na tle religii itp. ale nie rasizm. Rasizm, to jednak coś innego.
Swoją drogą, nie wykluczyłbym, że taka właśnie stereotypowa opinia stanie się wkrótce tak powszechna, że i ci podpadający rzeczywiście pod ten stereotyp muzułmanie przypomną sobie i doświadczą na własnej skórze, że niemrawa, multi-kulti itd europejska, czy też "zachodnia" cywlizacja, ma bogate tradycje wykluczania, segregowania a w końcu efekywnego i profesjonalnego mordowania.



Most reshares: 3

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2015-02-05 04:47:30 (1 comments, 3 reshares, 27 +1s)Open 

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Most plusones: 27

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2015-02-05 04:47:30 (1 comments, 3 reshares, 27 +1s)Open 

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2015-02-28 07:53:31 (0 comments, 0 reshares, 4 +1s)Open 

Strzał w sercu...

This is pretty cool: the authors attempted to prove the correctness of a common sorting implementation and discovered it was in fact flawed.  Seems an input has to be rather carefully crafted to trigger a failure, though.___Strzał w sercu...

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2015-02-27 13:42:53 (1 comments, 0 reshares, 24 +1s)Open 

Snow White Updated 
http://bit.ly/1Ew7n72

Snow White Updated 
http://bit.ly/1Ew7n72___

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2015-02-26 17:33:54 (0 comments, 1 reshares, 3 +1s)Open 

I thought I'd have a go at doing some unicode mathematics in G+. Sorry if you don't have all of ∂, ᵗ, ᵢ, ⱼ, ∑ and others in your font.

I highly recommend Boyd and Vanderberghe's book on convex optimisation: http://stanford.edu/~boyd/cvxbook/

Many of the exercises involve finding the duals of optimizations and I have to admit I quite enjoyed doing them. So I thought I'd make one up to entertain myself on the bus home. It's a Google bus so a sizable fraction of passengers has code or equations on their screen. I'm posting it here as it might be useful.

Hamilton's principle of least action casts classical dynamics as an optimisation problem so I thought I'd look at the derivation of Hamiltonian dynamics from Lagrangian dynamics.

First I'm going to discretise time so it's xᵢ instead of x(t). And I'm going toassume the ... more »

I thought I'd have a go at doing some unicode mathematics in G+. Sorry if you don't have all of ∂, ᵗ, ᵢ, ⱼ, ∑ and others in your font.

I highly recommend Boyd and Vanderberghe's book on convex optimisation: http://stanford.edu/~boyd/cvxbook/

Many of the exercises involve finding the duals of optimizations and I have to admit I quite enjoyed doing them. So I thought I'd make one up to entertain myself on the bus home. It's a Google bus so a sizable fraction of passengers has code or equations on their screen. I'm posting it here as it might be useful.

Hamilton's principle of least action casts classical dynamics as an optimisation problem so I thought I'd look at the derivation of Hamiltonian dynamics from Lagrangian dynamics.

First I'm going to discretise time so it's xᵢ instead of x(t). And I'm going to assume the Lagrangian is convex so we can use standard convex optimisation methods.

So here is a discretised least action problem.

(1) min(x) ∑ᵢ L(xᵢ, (Ax)ᵢ)

(That means minimizing with respect to all of the xᵢ)

In the usual version the sum is an integral and the second argument to L is the derivative of x with respect to time. Instead I'm using a linear operator A acting on the vector of xᵢs. I'll replace A with the derivative at the end.

The matrix A messes things up. It mixes together xᵢ and xⱼ for different i and j. If it didn't do this then we could just minimise each term in the sum individually.

We can get rid of the mixing simply by rewriting the problem as:

min(x, y) ∑ᵢ L(xᵢ, yᵢ)
such that y = Ax

The catch is that we've pushed the problem into the constraints. The objective does look more symmetrical now, so that's a nice bonus. Still, it'd be nice to get rid of the y or constraints somehow. Let's do a pretty standard optimisation thing. We'll look at the dual of the optimisation w.r.t. y, temporarily fixing x. Call the Lagrange multiplers pᵢ. That gives this unconstrained problem:

(2) min(y) ∑ᵢ L(xᵢ, yᵢ) + pᵢ((Ax)ᵢ-yᵢ)

We can pull a term out of the minimisation:

(min(y) ∑ᵢ L(xᵢ, yᵢ) - pᵢyᵢ) + p·Ax

The first part is a sum of completely independent terms. So we get

(∑ᵢ (min(yᵢ) L(xᵢ, yᵢ) - pᵢyᵢ)) + p·Ax

Let's define H(x, p) = max(y) (py - L(x, y)) (Notice the sign flip.)

So the solution to the dual problem is now:

(∑ᵢ -Hᵢ(xᵢ, pᵢ)) - p·Ax

Strong duality tells us that the maximum, with respect to p, of the dual problem is the minimum of the original problem. So a solution to problem (2) is given by:

max(p) (∑ᵢ -H(xᵢ, pᵢ)) - p·Ax

and the solution to the original problem (1) is

min(x) max(p) (∑ᵢ -H(xᵢ, pᵢ)) - p·Ax

That's it. That is now the original problem in Hamiltonian rather than Lagrangian form.

We can use calculus to deduce that at the optimum:

∂H(xᵢ, pᵢ)/∂pᵢ - (Ax)ᵢ = 0
∂H(xᵢ, pᵢ)/∂xᵢ - (Aᵗp)ᵢ = 0

Now we go to the continuum. x and p become functions of time. The matrix A becomes the derivative operator d/dt. Aᵗ becomes the adjoint -d/dt.

So we get Hamilton's equations:

∂H(x, p)/∂p = dx/dt
∂H(x, p)/∂x = -dp/dt

One thing that I learnt from this is that the minus sign in the second of these equations is coming from the adjoint of the derivative.

The other thing I learnt was that the entire derivation is completely natural from the point of optimisation. Unlike the usual accounts, I don't feel like I plucked random expressions out of a hat at any stage. I just grouped things together that seemed to go together.

By the way, the substitution of integrals for minima, products for sums and exponentials for linear functions turns the above into a (sketch of a) derivation of the Hamiltonian path integral from the Lagrangian path integral in quantum mechanics.

PS I composed all this using my vim script https://dl.dropboxusercontent.com/u/828035/math.vim___

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2015-02-25 18:17:47 (0 comments, 0 reshares, 6 +1s)Open 

Takiego mamy koleżkę (czy też koleżankę)

Did you know? Earth has another "moon". And it's got one "funky orbit"...

#space #moon #science #orbit #solarsystem #Cruithne___Takiego mamy koleżkę (czy też koleżankę)

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2015-02-23 17:57:43 (1 comments, 1 reshares, 3 +1s)Open 

Raczej nie do zobaczenia w "Cocomo" i produktach jego rozpadu...

Raczej nie do zobaczenia w "Cocomo" i produktach jego rozpadu...___

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2015-02-23 13:47:19 (0 comments, 0 reshares, 6 +1s)Open 

Największy ze zbudowanych przez naszych studentów dronów wznosi się 50 metrów nad ziemią i może odlecieć na 2 km od nadajnika. Ma rozpiętość skrzydeł 3,2 metra i podnosi ładunek kilkakrotnie cięższy od niego samego. Trzymamy kciuki za zespół JetStream na zawodach w Stanach Zjednoczonych:

http://www.tvn24.pl/wroclaw,44/dron-polskich-naukowcow-podbije-usa-i-powtorzy-sukces-sprzed-roku,517204.html

#politechnikawroclawska

Największy ze zbudowanych przez naszych studentów dronów wznosi się 50 metrów nad ziemią i może odlecieć na 2 km od nadajnika. Ma rozpiętość skrzydeł 3,2 metra i podnosi ładunek kilkakrotnie cięższy od niego samego. Trzymamy kciuki za zespół JetStream na zawodach w Stanach Zjednoczonych:

http://www.tvn24.pl/wroclaw,44/dron-polskich-naukowcow-podbije-usa-i-powtorzy-sukces-sprzed-roku,517204.html

#politechnikawroclawska___

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2015-02-13 07:58:10 (0 comments, 0 reshares, 5 +1s)Open 

Ciekawostka dla miłośników zabytków techniki. Niedawno wspominaliśmy burmistrza von Wintera, uznawanego za ojca współczesnych gdańskich wodociągów. Gdańskie Wodociągi były inwestycjami absolutnie pionierskimi – nowatorskimi w skali całej Europy. Co ważne, część obiektów zachowała się do tej pory. To świadectwo rozwoju technologicznego, a także rozwoju Gdańska. Dlatego warto je pielęgnować i kultywować ich historię. Właśnie temu służy projekt „Gdański Szlak Wodociągowy” prowadzony przez GIWK. Celem projektu  jest otwarcie ścieżki turystycznej, która umożliwi bliższe poznanie zachowanych obiektów oraz ich historii. Przyczynią się do tego na przykład ekspozycje tematyczne i atrakcje w zabytkowych (odpowiednio konserwowanych dla zachowania jak najlepszego stanu przez lata) obiektach wodociągowych, które będą w przyszłości udostępniane grupomzorganizowanym.
W pierwszej kolejności w zakres pro... more »

Ciekawostka dla miłośników zabytków techniki. Niedawno wspominaliśmy burmistrza von Wintera, uznawanego za ojca współczesnych gdańskich wodociągów. Gdańskie Wodociągi były inwestycjami absolutnie pionierskimi – nowatorskimi w skali całej Europy. Co ważne, część obiektów zachowała się do tej pory. To świadectwo rozwoju technologicznego, a także rozwoju Gdańska. Dlatego warto je pielęgnować i kultywować ich historię. Właśnie temu służy projekt „Gdański Szlak Wodociągowy” prowadzony przez GIWK. Celem projektu  jest otwarcie ścieżki turystycznej, która umożliwi bliższe poznanie zachowanych obiektów oraz ich historii. Przyczynią się do tego na przykład ekspozycje tematyczne i atrakcje w zabytkowych (odpowiednio konserwowanych dla zachowania jak najlepszego stanu przez lata) obiektach wodociągowych, które będą w przyszłości udostępniane grupom zorganizowanym.
W pierwszej kolejności w zakres projektu wejdą 3 obiekty:
•    zbiornik wody Stary Sobieski
•    zbiornik wody Stara Orunia
•    przepompownia ścieków Jelitkowo
Ponadto w ramach projektu udostępniony będzie taras widokowy na wieży ciśnień w Sobieszewie. Znajdzie się tam również ekspozycja, pozwalająca poznać historię i osobliwości przyrodnicze tego ciekawego rejonu Gdańska (szczegóły: http://blizejzrodel.pl/post/110164999584/wiezowy-zbiornik-wody-na-wyspie-sobieszewskiej).___

2015-02-06 14:52:33 (14 comments, 0 reshares, 14 +1s)Open 

Lubię, idąc przez miasto, zrównać krok z kimś spacerującym z psem i podsłuchać co też ów ktoś mówi swemu psu.Niektóre z psów z szczerością rozbrajającą ignorują te monologi, inne, przez grzeczność, udają zainteresowanie.

Lubię, idąc przez miasto, zrównać krok z kimś spacerującym z psem i podsłuchać co też ów ktoś mówi swemu psu.Niektóre z psów z szczerością rozbrajającą ignorują te monologi, inne, przez grzeczność, udają zainteresowanie.___

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2015-02-05 04:47:30 (1 comments, 3 reshares, 27 +1s)Open 

Stats

Stats___

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2015-02-01 17:57:53 (1 comments, 0 reshares, 3 +1s)Open 

This article in Science gives a good short rundown on the debate going on about the relationship between the American Mathematical Society and the NSA. For me personally, it's a frustrating read. (PDF)
http://www.sciencemag.org/content/347/6221/495.full.pdf

This article in Science gives a good short rundown on the debate going on about the relationship between the American Mathematical Society and the NSA. For me personally, it's a frustrating read. (PDF)
http://www.sciencemag.org/content/347/6221/495.full.pdf___

2015-01-31 18:41:23 (3 comments, 0 reshares, 10 +1s)Open 

Z notki o wczorajszym pożarze w Moskwie zamieszczonej w Onecie:
"Instytut Informacji Naukowej Nauk Społecznych to największy w Rosji ośrodek informacji naukowej w dziedzinie nauk społecznych i humanitarnych."
To chyba coś jak z tymi konwojami do Doniecka - mają tam jakieś nauki, które dla niepoznaki nazywają humanitarnymi... Ech, Onet...

Z notki o wczorajszym pożarze w Moskwie zamieszczonej w Onecie:
"Instytut Informacji Naukowej Nauk Społecznych to największy w Rosji ośrodek informacji naukowej w dziedzinie nauk społecznych i humanitarnych."
To chyba coś jak z tymi konwojami do Doniecka - mają tam jakieś nauki, które dla niepoznaki nazywają humanitarnymi... Ech, Onet...___

2015-01-28 07:02:33 (15 comments, 0 reshares, 8 +1s)Open 

Skończyłem w zeszłym tygodniu kolejną, po "Pierścieniach Saturna" i "Wyjechali", w mojej karierze czytelniczej książkę W.G.Sebalda, a mianowicie "Czuję. Zawrót głowy.". Jest coś w pisarstwie Sebalda, co spasowane wydaje mi się doskonale ze sposobem w jaki i ja podróżuję przez przestrzeń i czas i skrośnie do nich przez światy fikcji, światy wyobrażone lub przeżyte ale opisane przez kogoś innego. A piszę tak dlatego, że jeśli ktoś zmuszałby mnie do włożenia Sebalda w jakąś szufladkę, najmniej dyskomfortu odczuwałbym nazywając to co pisze literaturą drogi, gdzie droga wiedzie przez takie właśnie, skomplikowane w swojej naturze przestrzenie. 
W "Czuję." autor odbywa ruch kolisty. Wraca w podróży przez Włochy, Szwajcarię, Austrię Niemcy, w te same miejsca, w których już w ten czy inny sposób był,  podążając tropemwłasnym, ale trudnym do oddzielenia od innych tropów... more »

Skończyłem w zeszłym tygodniu kolejną, po "Pierścieniach Saturna" i "Wyjechali", w mojej karierze czytelniczej książkę W.G.Sebalda, a mianowicie "Czuję. Zawrót głowy.". Jest coś w pisarstwie Sebalda, co spasowane wydaje mi się doskonale ze sposobem w jaki i ja podróżuję przez przestrzeń i czas i skrośnie do nich przez światy fikcji, światy wyobrażone lub przeżyte ale opisane przez kogoś innego. A piszę tak dlatego, że jeśli ktoś zmuszałby mnie do włożenia Sebalda w jakąś szufladkę, najmniej dyskomfortu odczuwałbym nazywając to co pisze literaturą drogi, gdzie droga wiedzie przez takie właśnie, skomplikowane w swojej naturze przestrzenie. 
W "Czuję." autor odbywa ruch kolisty. Wraca w podróży przez Włochy, Szwajcarię, Austrię Niemcy, w te same miejsca, w których już w ten czy inny sposób był,  podążając tropem własnym, ale trudnym do oddzielenia od innych tropów. Zbierając bilety, rachunki, pojedyncze fotografie. Deptając sobie po piętach z Kafką czy Casanovą, walcząc z neurozami i wciąż natykając się na obrazy, znaki, czy inkarnacje postaci historycznych bądź fikcyjnych. 
Tak się składa, że dużo podróżuję, głównie tramwajami, po ciasnym okręgu, tropiąc codziennie siebie samego z wczoraj. Tak się składa, że nie są mi obce neurozy autora, oraz ja też w tych podróżach zbaczam w światy wykreowane i spotykam w tych tramwajach notorycznie najrozmaitsze postaci: od Luisa de Funesa, po wiotkie, chadzające w sztucznie podartych dżinsach modelki renesansowych malarzy.  Cóż więc dziwnego, że na kilkanaście dni znalazłem w osobie Sebalda towarzysza w tych podróżach? Towarzysza z którym z żalem się póki co rozstałem.___

2015-01-24 17:35:26 (6 comments, 0 reshares, 8 +1s)Open 

To miasto jest jak podstępny, złośliwy sadysta. Potrafi skrajnie wymiętego przez intensywny dzień pracy, sobotę dodajmy, człowieka, kiedy zdaje mu się, że nadchodzi błogi odpoczynek, zaskoczyć zerwaną trakcją tramwajową, koniecznością poruszania się autobusami, przesiadania się, obijania się o agresywnych szukających zaczepki pijaków i to wszystko w dniu najparszywszej, być może, póki co, tej zimy pogody.

To miasto jest jak podstępny, złośliwy sadysta. Potrafi skrajnie wymiętego przez intensywny dzień pracy, sobotę dodajmy, człowieka, kiedy zdaje mu się, że nadchodzi błogi odpoczynek, zaskoczyć zerwaną trakcją tramwajową, koniecznością poruszania się autobusami, przesiadania się, obijania się o agresywnych szukających zaczepki pijaków i to wszystko w dniu najparszywszej, być może, póki co, tej zimy pogody.___

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2015-01-24 17:21:22 (1 comments, 0 reshares, 4 +1s)Open 

:) "Figuring out what happened in a collider is like trying to figure out what your dog ate at the park yesterday. You can find out, but you have to sort through a lot of shit to do it."

How Three Guys With $10K and Decades-Old Data Almost Found the Higgs Boson First
An excellent Saturday physics read. Nice story.

#higgsboson   #lep   #lhc  ___:) "Figuring out what happened in a collider is like trying to figure out what your dog ate at the park yesterday. You can find out, but you have to sort through a lot of shit to do it."

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2015-01-22 05:46:47 (0 comments, 0 reshares, 5 +1s)Open 

Start Taking Notes By Hand Now! Here's why...

Start Taking Notes By Hand Now! Here's why...___

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2015-01-21 17:10:48 (1 comments, 0 reshares, 2 +1s)Open 

If you study the spherical harmonics without studying representation theory they are mysterious rabbits pulled out of hats. But if you study the representation theory of SO(3), their existence and many of their properties become obvious.

I've never managed to get my head around Bessel functions. To me they look like rabbits out of hats. Every time I read about them there's just some sequence of differential equations and Taylor series expansions that seem completely unmotivated. So what if they satisfy this or that differential equation, I could probably make up infinitely many other equations they satisfy. Who cares?

So it suddenly dawned on me that maybe Bessel functions carry a representation of some group and if I knew which group all would become clear. And then I thought this was pointless to think about because if they did carry a representation in a nice way it'd be... more »

If you study the spherical harmonics without studying representation theory they are mysterious rabbits pulled out of hats. But if you study the representation theory of SO(3), their existence and many of their properties become obvious.

I've never managed to get my head around Bessel functions. To me they look like rabbits out of hats. Every time I read about them there's just some sequence of differential equations and Taylor series expansions that seem completely unmotivated. So what if they satisfy this or that differential equation, I could probably make up infinitely many other equations they satisfy. Who cares?

So it suddenly dawned on me that maybe Bessel functions carry a representation of some group and if I knew which group all would become clear. And then I thought this was pointless to think about because if they did carry a representation in a nice way it'd be in every course on special functions and I'd have heard about it long ago. Unless there was a conspiracy to hide this fact from me.

So I tried doing a web search on Bessel functions and representation theory. And damn it! There' has been such a conspiracy! The Bessel functions carry a representation of the Lie algebra of the 2D Euclidean group in a natural way. But the only way to know about it is to read books that are overpriced and out of print. For some reason it's not taught much.

Oh...and there's this PhD thesis I found: http://arxiv.org/pdf/1309.2544v1.pdf

Just briefly: it's not really the Bessel functions that are important but the functions f_n(r,θ)=exp(inθ)J_n(r) thought of as a function on the plane expressed in polar coordinates. I've attached a contour plot of the real part of f_3. Just like you have with spherical harmonics (and the Hermite polynomials if you know the quantum harmonic oscillator) there are ladder operators relating the different f_n. Many of the properties of Bessel functions follow trivially from this. Some of the arbitrary looking differential differential equations follow when you look at how commutators and anticommutators of the ladder operators act on the representation.

What's more, there are many more special functions that carry representations of familiar groups. For example the hypergeometric functions are associated to SL(2,R).

I wish I'd known this many years ago.

I have to confess, it's still a bit rabbity to me. I've never studied the representation theory of the Euclidean groups so I wouldn't know, a priori, what to expect, whereas the "triangular" structure of the spherical harmonics is no surprise if you know SO(3).___

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2015-01-20 12:51:02 (3 comments, 1 reshares, 6 +1s)Open 

Zaplanuj podróż pociągiem w Mapach Google

Wejdźcie na stronę maps.google.pl lub uruchomcie aplikację Mapy Google i po kliknięciu ikony Wyznacz trasę, wybierzcie opcję Transport kolejowy (ikonę pociągu). Teraz musicie już tylko podać stację początkową i końcową, żeby dowiedzieć się jakim najlepszym połączeniem, realizowanym przez pociągi PKP Intercity (EIC Premium, EIC, IC oraz TLK), możecie pokonać tę trasę.

Więcej informacji o tej nowej funkcji znajdziecie, jak zawsze, na naszym blogu: goo.gl/ugbbE6

Zaplanuj podróż pociągiem w Mapach Google

Wejdźcie na stronę maps.google.pl lub uruchomcie aplikację Mapy Google i po kliknięciu ikony Wyznacz trasę, wybierzcie opcję Transport kolejowy (ikonę pociągu). Teraz musicie już tylko podać stację początkową i końcową, żeby dowiedzieć się jakim najlepszym połączeniem, realizowanym przez pociągi PKP Intercity (EIC Premium, EIC, IC oraz TLK), możecie pokonać tę trasę.

Więcej informacji o tej nowej funkcji znajdziecie, jak zawsze, na naszym blogu: goo.gl/ugbbE6___

2015-01-19 22:39:21 (2 comments, 0 reshares, 9 +1s)Open 

Na przywitanie przyznam się ze wstydem, że nie przeczytałem, ale wysłuchałem, bom zakupił promocyjnie w audiotece ś.p. Toma Clancy'ego i Marka Greaneya, co robotę po nieboszczyku dokończył, "Zwierzchnika". Z tym słuchaniem, to było tak, że miałem w okresie tuż po świętach poważne problemy z bezsennością, toteż nie chcąc umordować oczu, postanowiłem miast po nocach czytać, wysłuchać jakiejś lekkiej a długiej (antycypowałem bowiem dłuższe kłopoty ze snem) książki. Okazało się, że mimo wartkiej akcji w powieści, słuchanie mnie nieźle usypia, więc zwykle zapadałem w sen po kilku rozdziałach i budziłem się po kolejnych kilkunastu. Zajęło mi to słuchanie tedy dłużej niż pierwotnie sądziłem i liczbowo rzecz biorąc, przesłuchałem kiążkę kilka razy, głównie we śnie, ale skończyłem. Tyle o sobie, a teraz kilka słów o książce. No więcosnuta jest intryga wokół - ciary przechodzą - rosyjskiego ataku na U... more »

Na przywitanie przyznam się ze wstydem, że nie przeczytałem, ale wysłuchałem, bom zakupił promocyjnie w audiotece ś.p. Toma Clancy'ego i Marka Greaneya, co robotę po nieboszczyku dokończył, "Zwierzchnika". Z tym słuchaniem, to było tak, że miałem w okresie tuż po świętach poważne problemy z bezsennością, toteż nie chcąc umordować oczu, postanowiłem miast po nocach czytać, wysłuchać jakiejś lekkiej a długiej (antycypowałem bowiem dłuższe kłopoty ze snem) książki. Okazało się, że mimo wartkiej akcji w powieści, słuchanie mnie nieźle usypia, więc zwykle zapadałem w sen po kilku rozdziałach i budziłem się po kolejnych kilkunastu. Zajęło mi to słuchanie tedy dłużej niż pierwotnie sądziłem i liczbowo rzecz biorąc, przesłuchałem kiążkę kilka razy, głównie we śnie, ale skończyłem. Tyle o sobie, a teraz kilka słów o książce. No więc osnuta jest intryga wokół - ciary przechodzą - rosyjskiego ataku na Ukrainę (napisano, zanim Te Rzeczy Tam się podziały), zaszłości zimnowojennych, wielkich finansów itp. Sporo łubudu, czasem nieco bezwstydnie załopoce flaga USA, czasem brytyjska, są dłużyzny, ale w sumie warto. Ja bardzo rzadko ostatnio sięgam po taką literaturę, więc może trochę był dla mnie w tej lekturze efekt przypomnienia sobie zapomnianego smaku. W każdym razie - no, wyksztuszę to - podobała mi się. A zwłaszcza, ze względu na moją świeżo nabytą rusofobię, podobało mi się jak prażą do agresora z helikopterów hellfajerami ;)___

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2015-01-19 21:56:47 (1 comments, 1 reshares, 3 +1s)Open 

Partition and sum is fast

A “partition and sum” operation on a natural number is defined by treating the number as a long string of digits, inserting one or more plus signs in between some of the digits, and adding up the result. It is clear that this will produce a smaller number, and that doing this operation enough times will eventually terminate in a single digit answer. What is surprising is that if the partitions are chosen well, an arbitrarily large natural number in any number base can be reduced to a single digit in at most three iterations.

The recent paper Partition and Sum is Fast by Steve Butler, Ron Graham, and Richard Stong (http://arxiv.org/abs/1501.04067) contains an outline of a proof of this result; the full details are available in a paper by the same authors that is available from Ron Graham's web page.

The picturesho... more »

Partition and sum is fast

A “partition and sum” operation on a natural number is defined by treating the number as a long string of digits, inserting one or more plus signs in between some of the digits, and adding up the result. It is clear that this will produce a smaller number, and that doing this operation enough times will eventually terminate in a single digit answer. What is surprising is that if the partitions are chosen well, an arbitrarily large natural number in any number base can be reduced to a single digit in at most three iterations.

The recent paper Partition and Sum is Fast by Steve Butler, Ron Graham, and Richard Stong (http://arxiv.org/abs/1501.04067) contains an outline of a proof of this result; the full details are available in a paper by the same authors that is available from Ron Graham's web page.

The picture shows an example of an iterated “partition and sum” operation on the number 10211914 in base 10. The authors like this number because it is the birth date (in US format) of the popular mathematical writer Martin Gardner (1914–2010). By splitting the number up as 1+0+2+1+1+91+4, we obtain an answer of 100. Performing the reduction again, we obtain a single digit answer: 1+0+0=1.

There are other ways that the partition and sum operation could have been performed. The obvious way is simply to add up all the digits, which produces an answer of 19. It then takes two iterations to get from 19 to a single digit: 1+9=10 and 1+0=1. According to the authors, there is only one way to reduce 10211914 to a single digit in only two steps.

The reason the result is surprising is that the length of a number in base 10 is very close to its logarithm in base 10. The function log(x) grows very slowly with x, although it does tend to infinity as x tends to infinity. One might therefore think that extremely long numbers would take a large number of iterations to reduce to a single digit in this way, so it is surprising that only three iterations suffice, irrespective of the number of digits or the number base.

Many numbers only need two iterations. In base 2, every number can be reduced in two steps, and the same is true in base 3, with eleven exceptions. The exceptions, which need three iterations, are listed in the paper: the smallest is 1781 (2102222 in base 3) and the largest is 393901 (202000022221 in base 3).

Relevant links

The full version of the paper is called Collapsing numbers in bases 2, 3, and beyond and was written for “Gathering for Gardner 10”. It can be found at http://www.math.ucsd.edu/~ronspubs/mis_17_bases.pdf

Wikipedia on Martin Gardner: http://en.wikipedia.org/wiki/Martin_Gardner

#mathematics #scienceeveryday #spnetwork arXiv:1501.04067___

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2015-01-18 18:04:38 (8 comments, 1 reshares, 25 +1s)Open 

Mamy na G+ artystkę, ilustratorkę Andersena, kompletnie nie plusowaną jeszcze jak się okazuje...

Diabeł z tabakiery___Mamy na G+ artystkę, ilustratorkę Andersena, kompletnie nie plusowaną jeszcze jak się okazuje...

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2015-01-18 17:35:34 (4 comments, 0 reshares, 6 +1s)Open 

Sometimes, triangles just don't cut it.

Sometimes, triangles just don't cut it.___

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2015-01-14 08:35:24 (1 comments, 1 reshares, 17 +1s)Open 

#xkcd Heisenberg would be proud.

#xkcd Heisenberg would be proud.___

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2015-01-12 05:24:00 (0 comments, 0 reshares, 4 +1s)Open 

The fractional chromatic number of the plane

What is the smallest number of colours that it would take to colour the points of the plane in such a way that any two points at unit distance apart have different colours? It has been known since 1961 that this number is between 4 and 7, but even now, nobody knows what the answer is.

This question arises in the mathematical area known as graph theory. In combinatorics, a graph is a set of points, called vertices. Some of the vertices are are joined in pairs by edges, and two vertices joined in this way are said to be adjacent. The chromatic number of a graph is the smallest number of colours needed to colour the vertices of the graph in such a way that each pair of adjacent vertices has different colours.

If we reinterpret the word “adjacent” to mean that two points are exactly one unit of distance apart,the... more »

The fractional chromatic number of the plane

What is the smallest number of colours that it would take to colour the points of the plane in such a way that any two points at unit distance apart have different colours? It has been known since 1961 that this number is between 4 and 7, but even now, nobody knows what the answer is.

This question arises in the mathematical area known as graph theory. In combinatorics, a graph is a set of points, called vertices. Some of the vertices are are joined in pairs by edges, and two vertices joined in this way are said to be adjacent. The chromatic number of a graph is the smallest number of colours needed to colour the vertices of the graph in such a way that each pair of adjacent vertices has different colours.

If we reinterpret the word “adjacent” to mean that two points are exactly one unit of distance apart, then the number described in the first paragraph is the chromatic number of the plane, if we regard the points of the plane as the vertices of a graph.

To see why this number is at most 7, consider the grid of hexagons on the right of the picture, and imagine extending it to the whole plane in such a way that the diameter of each hexagon is slightly less than 1 unit. This produces a colouring of the vertices of the plane with 7 colours in such a way that every two points that are one unit apart will have different colours. The argument above establishes the upper bound of 7. The lower bound of 4 comes from a graph called the Moser spindle, which has 7 vertices and chromatic number of 4. The key point is that the Moser spindle can be drawn in the plane in such a way that all its edges have length 1.

An a:b colouring of a graph is a variant of these ideas, in which each vertex is assigned a subset of b colours out of a palette of a colours, in such a way that adjacent vertices are not allowed to have any of their colours in common. For example, the left half of the picture shows a 5:2 colouring of the dodecahedral graph. This consists of 20 vertices, corresponding to the 20 vertices of the dodecahedron, joined together with edges in the same way as the dodecahedron. In this case, each vertex is assigned two colours out of a palette of five (red, yellow, green, blue, magenta). The b-fold chromatic number of the graph is the smallest possible a such that the graph has an a:b colouring. 

If we take all the values of a to be the smallest possible, and let b tend to infinity, it can be proved using a short but nontrivial argument that the sequence a/b approaches a limit, called the fractional chromatic number of the graph. Furthermore, the sequence converges to its infimum (greatest lower bound), which means in particular that the fractional chromatic number is always less than or equal to the usual chromatic number. The colouring described in the preceding paragraph shows that the fractional chromatic number of the dodecahedral graph is at most 5/2=2.5.

The recent paper The Fractional Chromatic Number of the Plane by Daniel W. Cranston and Landon Rabern (http://arxiv.org/abs/1501.01647) makes progress on the problem of determining the fractional chromatic number of the plane. It was previously known that this number lies somewhere between 3.5556 and 4.3599, but the authors improve the lower bound to 76/21, which is about 3.6190.

Chromatic numbers and fractional chromatic numbers have applications to scheduling. Consider the 5:2 colouring of the dodecahedral graph shown in the picture. Suppose that each of the 20 vertices represents a committee that must meet for an hour, and that two vertices are joined by an edge if the corresponding committees cannot meet simultaneously, for example, if they share one or more members. Each of the five colours represents a meeting time period of half an hour; the “half” is because each vertex is assigned two colours. So long as the committees do not mind being interrupted half way through their meetings, the picture shows how it is possible for all the committees to meet in 5/2=2.5 hours without conflicts. The chromatic number of this graph can be shown to be 3, which means that it would take 3 hours for all the committees to meet if no interruptions were allowed.

Relevant links

The right half of the picture comes from the paper by Cranston and Rabern. The left half of the picture is by Claudio Rocchini, and appears on the Wikipedia page about fractional colouring:  http://en.wikipedia.org/wiki/Fractional_coloring

The Moser spindle: http://en.wikipedia.org/wiki/Moser_spindle

The book Fractional graph theory by Edward R. Scheinerman and Daniel H. Ullman is online (deliberately, I hope) at http://www.ams.jhu.edu/~ers/fgt/fgt.pdf. Appendix A.4 of the book gives the “short but nontrivial argument” mentioned earlier.

Cranston and Rabern mention that the well-known mathematician Paul Erdős (1913–1996) wrote in 1985 that he was “almost sure” that the chromatic number of the plane was strictly bigger than 4. Erdős also had some interesting ideas on what to do if aliens threatened to destroy the earth if we failed to tell them the values of certain Ramsey numbers. I have posted about this before at https://plus.google.com/101584889282878921052/posts/fgrMcGnauZb

#mathematics #sciencesunday  #spnetwork arXiv:1501.01647___

2015-01-10 18:32:21 (0 comments, 0 reshares, 2 +1s)Open 

W nowych Notices of AMS pierwsza część "wielogłosu" na temat Paula Erdosa i jego dzieła w stulecie urodzin. http://www.ams.org/notices/201502/rnoti-p121.pdf

W nowych Notices of AMS pierwsza część "wielogłosu" na temat Paula Erdosa i jego dzieła w stulecie urodzin. http://www.ams.org/notices/201502/rnoti-p121.pdf___

2015-01-10 09:41:13 (3 comments, 1 reshares, 17 +1s)Open 

Plan Londynu wydawnictwa A-Z Map Company Ltd., który pożyczyłem (i póki co nie oddałem) od brata, to istne arcydzieło kompresji. Sam plan zajmuje około 160 stron 440 - stronnicowej książeczki, która oprócz niego zawiera jeszcze rozbudowany indeks. Na niewielkich stronach upakowana jest ogromna ilość informacji, wszystkie najmniejsze nawet uliczki, ich nazwy, główne budynki, stacje metra itd. Skomplikowana natura miasta wymusza niezwykłe zabiegi edytorskie i typograficzne, zmienne wielkości czcionek, skróty, różne formy dzielenia i krzyżowania wyrazów aby móc zmieścić wszystkie nazwy. Przy tym, również dzięki mądremu stosowaniu koloru, całość jest czytelna i, o czym świadczy fakt, że posługując się tym planem bez trudu docierałem w wybrane wcześniej miejsca, nadaje się do praktycznego korzystania.
Lubię przeglądać i kartkować ten plan a dzisiaj zupełniezachwyciło mnie, że jako opis rzeki, przepły... more »

Plan Londynu wydawnictwa A-Z Map Company Ltd., który pożyczyłem (i póki co nie oddałem) od brata, to istne arcydzieło kompresji. Sam plan zajmuje około 160 stron 440 - stronnicowej książeczki, która oprócz niego zawiera jeszcze rozbudowany indeks. Na niewielkich stronach upakowana jest ogromna ilość informacji, wszystkie najmniejsze nawet uliczki, ich nazwy, główne budynki, stacje metra itd. Skomplikowana natura miasta wymusza niezwykłe zabiegi edytorskie i typograficzne, zmienne wielkości czcionek, skróty, różne formy dzielenia i krzyżowania wyrazów aby móc zmieścić wszystkie nazwy. Przy tym, również dzięki mądremu stosowaniu koloru, całość jest czytelna i, o czym świadczy fakt, że posługując się tym planem bez trudu docierałem w wybrane wcześniej miejsca, nadaje się do praktycznego korzystania.
Lubię przeglądać i kartkować ten plan a dzisiaj zupełnie zachwyciło mnie, że jako opis rzeki, przepływającej nb. przez park olimpijski w Stratford podano "RIVER LEA or LEE" pozostawiając otwartym jakiś toczący się spór, czy historyczną niejasność której autorzy nie czuli się na siłach rozstrzygać. I to mimo tego, że każda literka czy słowo w tym przedsięwzięciu zdaje się ważyć.
Pognałem do Google Maps by stwierdzić jak tam rozstrzygnięto ową kwestię. Lea. Przy okazji, w tym z kolei arcydziele nawigacji, kartografii cyfrowej, z automatycznymi wyszukiwaniami, zmienną ilością szczegółów w zależności od aktualnej skali, możliwością skoku do zdjęć, satelitarnych i zwykłych, czy zgoła podróżowania wirtualnego po wybranych ulicach, uderzyło mnie coś, właśnie dlatego, że tuż przedtem obcowałem z tym papierowym planem. Otóż mój (a właściwie brata) AZ, ma w sobie coś czego elektronicznym odpowiednikom brakuje. Jakąś nieuchwytną wibrację, którą dzieli z miastem, które przedstawia. Ujawnia gęstwinę, którą to miasto jest. Jego "wymiar fraktalny". Wzorzec, coś właściwe jemu. Może, jest tak, że tylko coś niedoskonałego, ograniczonego, walczącego bohatersko z medium w którym egzystuje może właściwie przedstawić na tym poziomie, gdzie odbieramy rzeczy jako całość, coś innego niedoskonałego, żyjącego i krzewiącego się na poły dziko, uwikłanego w swoją historię i w warunki geograficzne. Jak papierowy plan przedstawiający miasto.___

2015-01-09 22:08:17 (17 comments, 0 reshares, 19 +1s)Open 

No, bez jaj. Jeżeli ktoś wyraża zdanie, że wyznawcy islamu to potencjalni terroryści nienawidzący wartości jakim hołduje (bądź twierdzi, że hołduje) zachodnia cywilizacja, to nie oznacza, że jest rasistą. Można mu przypisać wiele - od w miarę łagodnych cech: ignorancji czy skłonności do nadmiernych uogólnień, po nienawiść na tle religii itp. ale nie rasizm. Rasizm, to jednak coś innego.
Swoją drogą, nie wykluczyłbym, że taka właśnie stereotypowa opinia stanie się wkrótce tak powszechna, że i ci podpadający rzeczywiście pod ten stereotyp muzułmanie przypomną sobie i doświadczą na własnej skórze, że niemrawa, multi-kulti itd europejska, czy też "zachodnia" cywlizacja, ma bogate tradycje wykluczania, segregowania a w końcu efekywnego i profesjonalnego mordowania.



No, bez jaj. Jeżeli ktoś wyraża zdanie, że wyznawcy islamu to potencjalni terroryści nienawidzący wartości jakim hołduje (bądź twierdzi, że hołduje) zachodnia cywilizacja, to nie oznacza, że jest rasistą. Można mu przypisać wiele - od w miarę łagodnych cech: ignorancji czy skłonności do nadmiernych uogólnień, po nienawiść na tle religii itp. ale nie rasizm. Rasizm, to jednak coś innego.
Swoją drogą, nie wykluczyłbym, że taka właśnie stereotypowa opinia stanie się wkrótce tak powszechna, że i ci podpadający rzeczywiście pod ten stereotyp muzułmanie przypomną sobie i doświadczą na własnej skórze, że niemrawa, multi-kulti itd europejska, czy też "zachodnia" cywlizacja, ma bogate tradycje wykluczania, segregowania a w końcu efekywnego i profesjonalnego mordowania.

___

2015-01-07 08:20:17 (4 comments, 0 reshares, 10 +1s)Open 

Zero kamuflażu. Z artykułu w podroze.onet.pl dotyczącego wyróżnienia, jako wartej zobaczenia atrakcji, Centrum Kongresowego w Krakowie w magazynie "Travel and Leisure':

"Centrum Kongresowe w Krakowie (ICE Kraków Congress Centre) to nowoczesne, wielofunkcyjne centrum kongresowe w Krakowie."

Zero kamuflażu. Z artykułu w podroze.onet.pl dotyczącego wyróżnienia, jako wartej zobaczenia atrakcji, Centrum Kongresowego w Krakowie w magazynie "Travel and Leisure':

"Centrum Kongresowe w Krakowie (ICE Kraków Congress Centre) to nowoczesne, wielofunkcyjne centrum kongresowe w Krakowie."___

2015-01-05 08:18:54 (7 comments, 0 reshares, 7 +1s)Open 

Z opisu wycieczki w Rainbow Tours:
"Stavanger to jednak nie tylko przemysł, ale i urzekająca starówka z białymi drewnianymi domkami z mini-ogrodkami pełnymi kwiatów i brukowanymi uliczkami wspinającymi się po uroczych pagórkach miasta. Jakby czas zatrzymał się tu na przełomie XVIII i XX w."
Jakaś enneakaidekafobia? 

Z opisu wycieczki w Rainbow Tours:
"Stavanger to jednak nie tylko przemysł, ale i urzekająca starówka z białymi drewnianymi domkami z mini-ogrodkami pełnymi kwiatów i brukowanymi uliczkami wspinającymi się po uroczych pagórkach miasta. Jakby czas zatrzymał się tu na przełomie XVIII i XX w."
Jakaś enneakaidekafobia? ___

2015-01-04 18:30:00 (17 comments, 0 reshares, 19 +1s)Open 

No i obejrzałem trzecią część Hobbita. Określiłbym ją jako dramat higieniczno-stomatologiczny z elementami brutalnego gwałtu na zasadach zachowania pędu, momentu pędu i prawach statyki budowli. Z różnych źródeł zapożyczeń rzucają się w oczy zwłaszcza Diuna i Cliffhanger z Stallone.

No i obejrzałem trzecią część Hobbita. Określiłbym ją jako dramat higieniczno-stomatologiczny z elementami brutalnego gwałtu na zasadach zachowania pędu, momentu pędu i prawach statyki budowli. Z różnych źródeł zapożyczeń rzucają się w oczy zwłaszcza Diuna i Cliffhanger z Stallone.___

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2015-01-03 23:03:43 (0 comments, 1 reshares, 7 +1s)Open 

Happy 123rd Birthday to J.R.R. Tolkien! Without him we would not have Hobbits to envy for their peaceful way of life, we would not have the word eucatastrophe, and there would not be a family of sea slugs called Smeagol.

Here is LotrProject's interactive map of Middle-earth, allowing you to explore the wonderful world he created, on the web.

P.S. If you want to know more about those sea slugs: http://lotrproject.com/blog/2014/03/23/9-things-related-to-tolkien-you-maybe-didnt-know/

Happy 123rd Birthday to J.R.R. Tolkien! Without him we would not have Hobbits to envy for their peaceful way of life, we would not have the word eucatastrophe, and there would not be a family of sea slugs called Smeagol.

Here is LotrProject's interactive map of Middle-earth, allowing you to explore the wonderful world he created, on the web.

P.S. If you want to know more about those sea slugs: http://lotrproject.com/blog/2014/03/23/9-things-related-to-tolkien-you-maybe-didnt-know/___

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2015-01-02 22:01:55 (0 comments, 0 reshares, 5 +1s)Open 

The mathematics of card shuffling

A faro shuffle is a technique for shuffling a deck of cards by cutting the deck exactly in half and riffling the two halves together so that they alternate perfectly. Performing a sequence of eight identical faro shuffles will return a standard deck of cards to its original order.

There are actually two types of faro shuffle, depending on the way in which the cards are interlaced in the last step. If the original top card stays on the outside after riffling (i.e., it stays on the top), then the shuffle is called an out-shuffle. On the other hand, if the top card moves one place down to the inside after riffling, then the shuffle is called an in-shuffle.

Some card magic tricks are based on a palindrome-like property of faro shuffles that is sometimes called stay stack. This means that two cards that are symmetrically... more »

The mathematics of card shuffling

A faro shuffle is a technique for shuffling a deck of cards by cutting the deck exactly in half and riffling the two halves together so that they alternate perfectly. Performing a sequence of eight identical faro shuffles will return a standard deck of cards to its original order.

There are actually two types of faro shuffle, depending on the way in which the cards are interlaced in the last step. If the original top card stays on the outside after riffling (i.e., it stays on the top), then the shuffle is called an out-shuffle. On the other hand, if the top card moves one place down to the inside after riffling, then the shuffle is called an in-shuffle.

Some card magic tricks are based on a palindrome-like property of faro shuffles that is sometimes called stay stack. This means that two cards that are symmetrically placed around the middle of the pack (for example, the third card from the top and the third card from the bottom) will stay symmetrically placed after a sequence of faro shuffles. This means that the order of the entire pack of cards after a sequence of faro shuffles can be determined from a knowledge of the top half of the pack together with a knowledge of the initial ordering.

Using a combination of out-shuffles and in-shuffles, a pack of cards can potentially be arranged in a large number of ways. Despite this, it is not possible to transform the deck into an arbitrary order using a combination of faro shuffles. Determining which arrangements can be reached using shuffles of various types is an interesting problem, which is investigated in the recent paper The mathematics of the flip and horseshoe shuffles by Steve Butler, Persi Diaconis and Ron Graham (http://www.arxiv.org/abs/1412.8533). As the title suggests, the authors are interested in shuffles other than the standard faro shuffles; the latter were studied in a 1983 paper by Diaconis, Graham, and William Kantor.

The flip shuffle is illustrated in the picture. It is the same as the faro shuffle except that one of the halves of the pack is turned upside down before being shuffled back in. As with the faro shuffle, the flip shuffle comes in “out” and “in” versions. The picture illustrates the “out” version, if we assume that all the cards were face up to begin with. Theorem 2 of the paper explains how if n is a positive integer, then the combinatorics of shuffling a deck of 2n cards with flip shuffles is essentially the same as that of shuffling a deck of 4n cards with the usual faro shuffles. For example, if we take n=13, shuffling a deck of 26 cards with flip shuffles is very similar to shuffling a standard deck of 52 cards with faro shuffles. “Very similar” means not only that the number of possible arrangements of cards is the same in each case, but also that the groups of symmetries that arise are essentially the same (“canonically isomorphic”).

The reason for this close similarity has to do with the “stay stack” property. An obvious way to index a deck of cards is by the numbers 1, 2, 3, ..., 52,  where 1 refers to the top card, but it turns out to be useful instead to index the cards with the numbers 1, 2, 3, ..., 26, –26, –25, –24, ..., –1, where 1 is the top card and –1 is the bottom card. Now imagine shuffling the 52 cards using faro shuffles, then cutting the pack and selecting the top half. Within the 26 top cards, we can then replace each a card with a negative index (say, –11) with a flipped-over version of the corresponding positively indexed card (with index 11). Theorem 2 is proved by using this identification.

The part of the paper by Butler, Diaconis and Graham that deals with horseshoe shuffles is particularly interesting. A horseshoe shuffle is the same as a flip shuffle, except that every card remains facing the correct way at every step of the shuffle. Like the other shuffles, a horseshoe shuffle comes in “out” and “in” varieties. Theorem 3 of the paper lists the possible groups of symmetries arising from horseshoe shuffles on a deck of 2n cards, where n is a positive integer. If n is odd (and not equal to 3) then the group is the full symmetric group on 2n objects; this means that in this case, the pack can be mixed in any way you like using a combination of horseshoe shuffles. If n is even (and not equal to 6 or to a power of 2) then the group is the alternating group on 2n objects; this means that it is possible to reach exactly half of the possible permutations of the deck using horseshoe shuffles.

The authors admit to being annoyed at not being able to identify the symmetry group in general in the case where n is a power of 2. However, if n=3, then the group turns out to be the symmetric group on 5 objects, and if n=6, something surprising happens: the symmetry group is the Mathieu group M_12. The group M_12 is a famous example of a simple group. Groups are mathematical objects that measure symmetry in the same way that numbers measure quantity, and simple groups play the role in group theory that prime numbers play in arithmetic. With 26 exceptions, all finite simple groups fit into 18 infinite families. 

The group M_12, which consists of 95040 symmetries, is the second smallest of the 26 exceptional groups. This means that a deck of 12 cards can be horseshoe shuffled to exactly 95040 (8x9x10x11x12) different configurations. The group M_12 has the property of being sharply 5-transitive, which means that it is possible to use horseshoe shuffles to shuffle your five favourite cards in a deck of 12 to your five favourite positions, in any order you like (“5-transitive”), but also that the position of the other seven cards is then completely fixed (“sharply”).

Relevant links

The Mathieu groups: http://en.wikipedia.org/wiki/Mathieu_group

Faro shuffle: http://en.wikipedia.org/wiki/Faro_shuffle

The largest of the 26 sporadic finite simple groups is the Monster, which is discussed in my post here: https://plus.google.com/101584889282878921052/posts/9sKMLRJYjna

The picture here appears in the paper by Butler, Diaconis and Graham, but I do not know who the photographer is.

#mathematics #scienceeveryday #spnetwork arXiv:1412.8533___

2014-12-31 23:42:49 (2 comments, 0 reshares, 15 +1s)Open 

W tv pokazali jakiś czas temu fajerwerki z Moskwy. Całkiem skromne bym powiedział. Uczułem coś na kształt wdzięczności niemal, bo plany w 2014 Rosjanie rozważali ambitne: od pokazowego walnięcia atomowego nad bagnami nad Prypecią po takież samo nad Warszawą. Miejmy nadzieję, że i w nadchodzącym roku obejdą się straszeniem.

W tv pokazali jakiś czas temu fajerwerki z Moskwy. Całkiem skromne bym powiedział. Uczułem coś na kształt wdzięczności niemal, bo plany w 2014 Rosjanie rozważali ambitne: od pokazowego walnięcia atomowego nad bagnami nad Prypecią po takież samo nad Warszawą. Miejmy nadzieję, że i w nadchodzącym roku obejdą się straszeniem.___

2014-12-31 22:53:17 (1 comments, 0 reshares, 3 +1s)Open 

No, do zawodów 2014 zaliczam niniejszym American Mathematical Monthly, który nie wydał póki co numeru grudniowego (a w listopadowym, tematycznym, poświęconym biologii matematycznej i matematyce w biologii, nie zamieścił działu "Problems"). A listy rozpaczliwe by przedłużyć subskrypcję słali regularnie...

No, do zawodów 2014 zaliczam niniejszym American Mathematical Monthly, który nie wydał póki co numeru grudniowego (a w listopadowym, tematycznym, poświęconym biologii matematycznej i matematyce w biologii, nie zamieścił działu "Problems"). A listy rozpaczliwe by przedłużyć subskrypcję słali regularnie...___

2014-12-31 21:10:18 (2 comments, 0 reshares, 3 +1s)Open 

Naliczanie roczne kończy się zawsze jakimś absurdalnie gigantycznym rachunkiem trudnym do spłacenia. Trzeba przejść na naliczanie sekundowe. 

Naliczanie roczne kończy się zawsze jakimś absurdalnie gigantycznym rachunkiem trudnym do spłacenia. Trzeba przejść na naliczanie sekundowe. ___

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2014-12-26 21:46:42 (1 comments, 0 reshares, 10 +1s)Open 

I recently read Liskov and Zilles' paper Programming with Abstract Data Types from 1974 (I think) which you can find here: http://www.cs.iastate.edu/~hridesh/teaching/362/07/01/papers/p50-liskov.pdf

It seems to me this is one of the most important papers in computer science and it reminded me of the moment that I think is one of the most important in a pure mathematician's education.

When I was an undergraduate (doing the Cambridge Tripos) we had a Rings and Modules course and met the concept of the tensor product. Up until this point, every mathematical object had been defined pretty much the same way. It was a set, and we'd get a description of its elements. For example, the product of two groups, A×B, consists of the set of pairs of the form (a,b) with a form A and b from B, and so on.

But the definition of tensor product wasn't like this. It used thed... more »

I recently read Liskov and Zilles' paper Programming with Abstract Data Types from 1974 (I think) which you can find here: http://www.cs.iastate.edu/~hridesh/teaching/362/07/01/papers/p50-liskov.pdf

It seems to me this is one of the most important papers in computer science and it reminded me of the moment that I think is one of the most important in a pure mathematician's education.

When I was an undergraduate (doing the Cambridge Tripos) we had a Rings and Modules course and met the concept of the tensor product. Up until this point, every mathematical object had been defined pretty much the same way. It was a set, and we'd get a description of its elements. For example, the product of two groups, A×B, consists of the set of pairs of the form (a,b) with a form A and b from B, and so on.

But the definition of tensor product wasn't like this. It used the diagram attached and specified the product, not in terms of what the elements were made of, but in terms of some functions mapping to and from the product. I, like many of my colleagues, had a really hard time with this. (Me especially so, because at this time I was more of an applied mathematician with ambitions in physics.) It felt a bit like trying to bake a cake by being told the effect it was to have on its consumers.  How are you supposed to make it without knowing what goes inside? I managed to do the problems by seeing enough specific examples of tensor products to get a feel for them, but I didn't, at the time, fully grok the rationale of defining things this way.

Years later I was learning C++. I got to the bit where the private keyword is introduced. At first I thought "well, that's useful. You can stop people accidentally messing with the internal workings of your classes". Only a lot later did it dawn on me that the private keyword does something much more profound. It allows you to define classes much like the definition of the tensor product. Sometimes you don't want it to matter what the ingredients of a C++ class are. You just want to specify the functions you can use it with. And that's what private does: it enables abstraction by making the internals irrelevant, exactly as described in Liskov and Zilles. I already had a PhD in mathematics at that time. But strangely, seeing abstraction used in the context of C++ allowed me to abstract the notion of abstraction itself.

The fact that the two kinds of abstraction aren't taught together seems like a lost opportunity to me. I came out of my mathematics education thinking that computing met mathematics only in numerical methods, and maybe a bit of number theory. I was completely ignorant of the fact that mathematics, and in particular abstract algebra, also had much to say about the structure of computer programs.___

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2014-12-25 19:18:48 (1 comments, 0 reshares, 7 +1s)Open 

Tym co pracują dziś...

Alone in the lab?
http://www.phdcomics.com/comics.php?f=1773___Tym co pracują dziś...

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2014-12-19 03:39:16 (4 comments, 1 reshares, 13 +1s)Open 

Extreme Pair Programming
Source: http://redd.it/2pofnu

Extreme Pair Programming
Source: http://redd.it/2pofnu___

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2014-12-15 12:55:07 (2 comments, 1 reshares, 8 +1s)Open 

fioletowi...

fioletowi...___

2014-12-14 21:26:21 (2 comments, 1 reshares, 12 +1s)Open 

Wróciłem z "Interstellar". Z mieszanymi uczuciami, choć ogólnie warto. Straszne pomieszanie czasem fabularnie a czasem wizualnie różnych filmów jakie już widziałem np. "Odysei kosmicznej" czy "Grawitacji". Film generalnie o tematyce katastroficzno-relatywistyczno-topologicznej osnuty wokół dwóch problemów natury moralnej i posiadający kilka przesłań, w tym niektóre (ale nie wszystkie) dość naiwne. Realizatorzy dzielnie choć ze zmiennym szczęściem walczyli by zwizualizować nam gwałtownie zmienne tensory metryczne albo jak jawiłby się nam świat gdyby nasze 3-wymiarowe doświadczenie wizualne przenieść do 4-wymiarowej powierzchni jakoś inaczej położonej w 5-pięcio, ich zdaniem, wymiarowym świecie niż ta którą odbieramy jako nasz świat.
Wszystko to w ok. 3 godziny w przyzwoitej cenie.


Wróciłem z "Interstellar". Z mieszanymi uczuciami, choć ogólnie warto. Straszne pomieszanie czasem fabularnie a czasem wizualnie różnych filmów jakie już widziałem np. "Odysei kosmicznej" czy "Grawitacji". Film generalnie o tematyce katastroficzno-relatywistyczno-topologicznej osnuty wokół dwóch problemów natury moralnej i posiadający kilka przesłań, w tym niektóre (ale nie wszystkie) dość naiwne. Realizatorzy dzielnie choć ze zmiennym szczęściem walczyli by zwizualizować nam gwałtownie zmienne tensory metryczne albo jak jawiłby się nam świat gdyby nasze 3-wymiarowe doświadczenie wizualne przenieść do 4-wymiarowej powierzchni jakoś inaczej położonej w 5-pięcio, ich zdaniem, wymiarowym świecie niż ta którą odbieramy jako nasz świat.
Wszystko to w ok. 3 godziny w przyzwoitej cenie.
___

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2014-12-12 20:07:36 (5 comments, 1 reshares, 10 +1s)Open 

Gorzka pigułeczka.



Gorzka pigułeczka.

___

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2014-12-10 19:48:26 (0 comments, 0 reshares, 8 +1s)Open 

Did you see the stunning dress worn by May Britt Moser at the Nobel Prize ceremony today? It represents her work on grid neurons. The dress is pieced together as a grid and the beading is in the shape of three large neurons. As the light catches the beading, it sparkles as if electricity is coursing through the nerve fibers! 

I interview the designer, +matthew hubble at +Scientific American http://blogs.scientificamerican.com/psi-vid/2014/12/10/glittering-nobel-gown-represents-scientists-work/

Did you see the stunning dress worn by May Britt Moser at the Nobel Prize ceremony today? It represents her work on grid neurons. The dress is pieced together as a grid and the beading is in the shape of three large neurons. As the light catches the beading, it sparkles as if electricity is coursing through the nerve fibers! 

I interview the designer, +matthew hubble at +Scientific American http://blogs.scientificamerican.com/psi-vid/2014/12/10/glittering-nobel-gown-represents-scientists-work/___

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2014-12-09 20:55:10 (1 comments, 2 reshares, 21 +1s)Open 

Żart intertekstualny.

#Hobbit #GoT___Żart intertekstualny.

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2014-12-09 17:11:39 (2 comments, 0 reshares, 10 +1s)Open 

Pięknym morałem z tego jest: nawet jeśliś stary i wolny wciąż może cię czekać przygoda!

Doesn't need an explanation if you ask me, but ...   

Robustness - “The one thing we really like about this computer is that it doesn’t get destroyed by radiation,” said Matt Lemke, NASA’s deputy manager for Orion’s avionics, power and software team, in the report. “It can be upset, but it won’t fail. We’ve done a lot of testing on the different parts of the computer. When it sees radiation, it might have to reset, but it will come back up and work again.”

Proven Technology - The computer is based on a well-tested Honeywell system used in 787 jetliners. 

Cost - The use of commercial off the shelf technology overall reduces the risk and cost for a mission, especially when systems are built in triplicate for redundancy.  Also, human expertise with the product is usually readily available and therefore less expensive.  

...___Pięknym morałem z tego jest: nawet jeśliś stary i wolny wciąż może cię czekać przygoda!

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2014-12-09 04:24:31 (0 comments, 0 reshares, 7 +1s)Open 

___

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2014-12-07 19:43:24 (0 comments, 0 reshares, 8 +1s)Open 

Apropos szeroko reklamowanej wystawy (na której nie byłem) bardzo ciekawa rozmowa w Wyborczej, lekko przyprawiona naszym tu krakowskim narcyzmem.
http://m.krakow.gazeta.pl/krakow/1,106511,17082869,Prof__Jacek_Purchla__Cywilizacja_przyszla_z_Galicji.html?utm_source=m.gazeta.pl&utm_medium=testbox&utm_campaign=lokale

Apropos szeroko reklamowanej wystawy (na której nie byłem) bardzo ciekawa rozmowa w Wyborczej, lekko przyprawiona naszym tu krakowskim narcyzmem.
http://m.krakow.gazeta.pl/krakow/1,106511,17082869,Prof__Jacek_Purchla__Cywilizacja_przyszla_z_Galicji.html?utm_source=m.gazeta.pl&utm_medium=testbox&utm_campaign=lokale___

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2014-12-07 11:30:54 (5 comments, 3 reshares, 16 +1s)Open 

Vintage http://physicsbuzz.physicscentral.com/2014/12/the-29-best-retro-photos-from-cerns.html

Vintage http://physicsbuzz.physicscentral.com/2014/12/the-29-best-retro-photos-from-cerns.html___

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2014-12-06 20:23:51 (0 comments, 0 reshares, 14 +1s)Open 

___

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2014-12-06 18:55:13 (0 comments, 0 reshares, 5 +1s)Open 

Ramanujan's nested radical

In 1911, the mathematician Srinivasa Ramanujan (1887–1920) posed a problem in the Journal of the Indian Mathematical Society: to find the value of the radical expression √(1 + 2√(1 + 3√(1 + ...))). After six months, nobody had been able to solve the problem, so Ramanujan revealed the surprisingly simple answer, which is 3.

One strategy that mathematicians sometimes use when they are struggling to solve a problem is to generalize the problem. This sounds like a counterintuitive thing to do, but it is useful in this situation, as follows.

Define the function f(x) = x + n + a, where n and a are numbers. It can be easily checked that f(x)^2 = ax + (n+a)^2 + x(f(x+n)), because both sides of the equation are equal to x^2 + n^2 + a^2 + 2an + 2ax + 2nx. Taking square roots of each side shows that f(x) = √(ax + (n+a)^2 +xf(x+n)).... more »

Ramanujan's nested radical

In 1911, the mathematician Srinivasa Ramanujan (1887–1920) posed a problem in the Journal of the Indian Mathematical Society: to find the value of the radical expression √(1 + 2√(1 + 3√(1 + ...))). After six months, nobody had been able to solve the problem, so Ramanujan revealed the surprisingly simple answer, which is 3.

One strategy that mathematicians sometimes use when they are struggling to solve a problem is to generalize the problem. This sounds like a counterintuitive thing to do, but it is useful in this situation, as follows.

Define the function f(x) = x + n + a, where n and a are numbers. It can be easily checked that f(x)^2 = ax + (n+a)^2 + x(f(x+n)), because both sides of the equation are equal to x^2 + n^2 + a^2 + 2an + 2ax + 2nx. Taking square roots of each side shows that f(x) = √(ax + (n+a)^2 + xf(x+n)). This expression also gives a formula for f(x+n), which can then be substituted into the right hand side of the expression for f(x). Doing this recursively and then setting x=2, n=1 and a=0 gives the required identity, because f(2)=3.

This sketch of a proof has some loose ends. For example, the square roots are intended to be positive square roots, and it is not completely clear that the infinite expression given converges to 3 in any reasonable sense. However, the convergence turns out to be fast, as +John Cook demonstrated in a blog post from last year: http://www.johndcook.com/blog/2013/09/13/ramanujans-nested-radical/

In the same 1911 journal contribution, Ramanujan asked another similar question about nested radicals. This turned out to be the case x=2, n=1, a=1 of the same identity, which results in a nested radical expression that evaluates to 4. Ramanujan made a total of 58 such contributions to the Journal of the Indian Mathematical Society, including the famous Rogers–Ramanujan identities.

Relevant links
The 2005 paper On Ramanujan's nested roots expansion by K. Srinivasa Rao and G. Vanden Berghe explores this identity and some related results in detail: http://zariski.files.wordpress.com/2010/05/sr_nroots.pdf

Wikipedia on nested radicals: http://en.wikipedia.org/wiki/Nested_radical

Wikipedia on the Rogers–Ramanujan identities: http://en.wikipedia.org/wiki/Rogers–Ramanujan_identities

My post from Pi Day about Ramanujan's approximation to π: https://plus.google.com/101584889282878921052/posts/74oomcTuJoV

(Found via Cliff Pickover on Twitter.)

#mathematics #scienceeveryday  ___

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2014-12-05 18:26:40 (0 comments, 0 reshares, 1 +1s)Open 

Good Morning C++:

  Some great FREE content for the community from Effective C++, Third Edition: 55 Specific Ways to Improve Your Programs and Design by Scott Meyers

Link: http://ptgmedia.pearsoncmg.com/images/9780321334879/samplepages/0321334876.pdf

  Content Excerpt:

  "Clearly a BuyTransaction constructor will be called, but first, a Transaction constructor must be called; base class parts of derived class objects are constructed before derived class parts are. The last line of the Transaction constructor calls the virtual function logTransaction, but this is where the surprise comes in. The version of logTransaction that’s called is the one in Transaction, not the one in BuyTransaction — even though the type of object being created is BuyTransaction. During base class construction, virtual functions never go down into derived classes. Instead, theobject ... more »

Good Morning C++:

  Some great FREE content for the community from Effective C++, Third Edition: 55 Specific Ways to Improve Your Programs and Design by Scott Meyers

Link: http://ptgmedia.pearsoncmg.com/images/9780321334879/samplepages/0321334876.pdf

  Content Excerpt:

  "Clearly a BuyTransaction constructor will be called, but first, a Transaction constructor must be called; base class parts of derived class objects are constructed before derived class parts are. The last line of the Transaction constructor calls the virtual function logTransaction, but this is where the surprise comes in. The version of logTransaction that’s called is the one in Transaction, not the one in BuyTransaction — even though the type of object being created is BuyTransaction. During base class construction, virtual functions never go down into derived classes. Instead, the object behaves as if it were of the base type. Informally speaking, during base class construction, virtual functions aren’t."

"There’s a good reason for this seemingly counterintuitive behavior.
Because base class constructors execute before derived class constructors, derived class data members have not been initialized when base class constructors run. If virtual functions called during base class construction went down to derived classes, the derived class functions would almost certainly refer to local data members, but those data members would not yet have been initialized. That would be a non-stop ticket to undefined behavior and late-night debugging sessions. Calling down to parts of an object that have not yet been initialized is inherently dangerous, so C++ gives you no way to do it."

"It’s actually more fundamental than that."

#CPlusPlus   #CPP  ___

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