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www.abingdon.org.uk 9 Princeton Maths Challenge – superb performance! Congratulations to our team of mathematicians who came 3rd out of all postal entries worldwide and 11th overall in the Princeton University Maths Challenge ‘Power Round’: a set of undergraduate level mathematics problems set by the students of Princeton University, this year on Lie Algebras. Abingdon’s team was organised and led by Ben Wilson with James Hogge, Sean Kao and Hayden Ramm. Fifth year physicists visited the Rutherford Appleton Laboratory at Harwell for a Talk Physics lecture on “Pressing FIRE on the most powerful laser in the world!”. Dr Ceri Brenner from RAL spoke about future research projects and current research on utilising lasers for different applications including nuclear fusion. Abingdon News These are just a few of the comments we received from the first year boys at the end of their first week at Abingdon: “ Before my first day, I felt nervous and found the whole prospect of secondary school a bit daunting. After this first week, all my worries have evaporated! “ I could hardly believe my eyes when I saw how many activities we could choose from in the Other Half. “ My favourite lesson so far has been Geography because I was constantly laughing my head off! “ There are so many activities in sports and I loved my PE lessons! I am really looking forward to doing drama in the AMAZING Amey Theatre! I am also excited for DT with all the new equipment to use. The science block is AMAZING. “ Although I’m one of the smallest fish in the sea, I now feel quite confident because I’ve learnt where each classroom is and what time to be there. First Years’ First Week Talk Physics 5.12 Problem 5.3.1 As in the worked example, e i,j denotes then N × Nmatrix with entry 1 at co- ordinates i,j and zeros elsewhere. (i) L = so 2 n +1 . We know from Problem 2.2.2 that any element x can be written in the form v i =   0 B C − C T P Q − B T R − P T   where Q and R are equal to their negative transposes, P,Q,R are n × n , and B,C are n × 1. H is the set of diagonal matrices in L, so the elements of H can all be written n i =1 c i ( e i +1 ,i +1 − e i +1 ,i + n +1 ) where c i ∈ C . The remainder of basis elements are written p i,j = e i +1 ,j +1 − e j + n +1 ,i + n +1 where i = j and 1 ≤ i,j ≤ n , q i,j = e i +1 ,j + n +1 − e j +1 ,i + n +1 where 1 ≤ i < j ≤ n , any p T i,j , b j = e 1 ,j +1 − e j + n +1 , 1 , and any b T j . For an arbitrary h ∈ H , [ h,p i,j ] = ( c i − c j ) p i,j [ h,q i,j ] = ( c i + c j ) q i,j [ h,q T i,j ] = − ( c i + c j ) q T i,j [ h,b j ] = c j b j [ h,b T j ] = c j b T j so our roots are root eigenspace λ i − λ j span { p i,j ,p j,i } λ i + λ j span { q i,j ,q j,i } λ j span { b i,j ,b j,i } For H ∗ , a vector space with basis λ i , we see that { λ 1 − λ 2 ,λ 2 − λ 3 ,...,λ n − 1 − λ n ,λ n } is a base, which we can use to build a Dynkin diagram. Computation gives that λ i − λ i +1 ,λ j − λ j +1 =   2 if i = j 1 if | i − j | = 1 0 otherwise λ i − λ i +1 ,λ n = − 1 √ 2 if i = n − 1 0 otherwise A very small part of the solution! Abingdon has joined the new digital market place for parents. School Notices enables parents to communicate, advertise and trade between themselves and parents at other independent schools, whilst also benefiting some of our school causes through the sharing of advertising income. Further details: www.schoolnotices.co.uk and register at: schoolnotices.co.uk/abingdon-school