98 research outputs found
On the classification of easy quantum groups
In 2009, Banica and Speicher began to study the compact quantum subgroups of
the free orthogonal quantum group containing the symmetric group S_n. They
focused on those whose intertwiner spaces are induced by some partitions. These
so-called easy quantum groups have a deep connection to combinatorics. We
continue their work on classifying these objects introducing some new examples
of easy quantum groups. In particular, we show that the six easy groups O_n,
S_n, H_n, B_n, S_n' and B_n' split into seven cases on the side of free easy
quantum groups. Also, we give a complete classification in the half-liberated
case.Comment: 39 pages; appeared in Advances in Mathematics, Vol. 245, pages
500-533, 201
Decompositions of complete uniform hypergraphs into Hamilton Berge cycles
In 1973 Bermond, Germa, Heydemann and Sotteau conjectured that if divides
, then the complete -uniform hypergraph on vertices has a
decomposition into Hamilton Berge cycles. Here a Berge cycle consists of an
alternating sequence of distinct vertices and
distinct edges so that each contains and . So the
divisibility condition is clearly necessary. In this note, we prove that the
conjecture holds whenever and . Our argument is based on
the Kruskal-Katona theorem. The case when was already solved by Verrall,
building on results of Bermond
Gauss decomposition for Chevalley groups, revisited
In the 1960's Noboru Iwahori and Hideya Matsumoto, Eiichi Abe and Kazuo
Suzuki, and Michael Stein discovered that Chevalley groups over a
semilocal ring admit remarkable Gauss decomposition , where
is a split maximal torus, whereas and
are unipotent radicals of two opposite Borel subgroups
and containing . It follows from the
classical work of Hyman Bass and Michael Stein that for classical groups Gauss
decomposition holds under weaker assumptions such as \sr(R)=1 or \asr(R)=1.
Later the second author noticed that condition \sr(R)=1 is necessary for
Gauss decomposition. Here, we show that a slight variation of Tavgen's rank
reduction theorem implies that for the elementary group condition
\sr(R)=1 is also sufficient for Gauss decomposition. In other words,
, where . This surprising result shows that
stronger conditions on the ground ring, such as being semi-local, \asr(R)=1,
\sr(R,\Lambda)=1, etc., were only needed to guarantee that for simply
connected groups , rather than to verify the Gauss decomposition itself
WASSTONEHENGEANOBSERVATORY?
pointedexactlytothedirectionofsunriseatsolstice.Severalsimilarcoincidenceswerediscovered,whichleadmanypeopletobelievethatStonehengewasanancientobservatory.Inthispaper,weanalyzethisproblemfromageometricviewpointandshowthatsimilar coincidencescanhappen(withaprobability)forrandomlyplaced stones.Thus,Stonehenge'scoincidencescannotbeviewedasproof thatthisplacewasanobservatory. In1965,G.S.HawkinsandJ.B.Whitepublishedabook(Hawkins etal1965)inwhichtheyarguedthatStonehenge,amysterious 1.HAWKINS:ITWAS ancientthree-circlestonestructureinEngland,wasactuallyan ancientobservatory.Forexample,theydiscoveredthatadirection formedbytwoof60stonesoftheoutercirclepoints(withahigh accuracy,lessthan1)wasthedirectioninwhichthesunrises duringthesolarsolstice(actually,tothedirectioninwhichsun rose placetwostonesatrandom,thentheprobabilitythatthesetwo stonespointinthisdirectionisminuscule. 4;000yearsago,whenStonehengewasbuilt).Ifwesimply tweenthestonescouldbeinterpretedinreasonableastronomicaltheauthorsusedacomputertocheckifanyotherdirectionsbe-terms,andtheyindeedfoundmanysuchdirections. Toconvincethosewhowerenotconvincedbythiscoincidence, arethankfultoMariaBeltranforvaluablecomments
Automated Cluster-Based Web Service Performance Tuning
system to improve the performance of a cluster-based web service system. The performance improvement cannot easily be achieved by tuning individual components for such a system. The experimental results show that there is no single configuration for the system that performs well for all kinds of workloads. By tuning the parameters, the Active Harmony helps the system adapt to different workloads and improve the performance up to 16%. For scalability, we demonstrate how to reduce the time when tuning a large system with many tunable parameters. Finally an algorithm is proposed to automatically adjust the structure of cluster-based web systems, and the system throughput is improved up to 70 % using this technology. I
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